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Expand the square in the first one:","paragraph-hpW4hX80L",{"__TSPROSE_proseElement":215,"schema":483,"data":484,"storageKey":486,"id":487},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":485,"freeze":219},"(m+n)^2 - m^2 - n^2 = \\cancel{m^2} + 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = 2mn","$$ (m+n)^2 - m^2 - n^2 = \\cancel{m^2} + 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = 2mn $$","blockMath-tPacNDWyD",{"__TSPROSE_proseElement":215,"schema":489,"children":490,"id":494},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[491],{"__TSPROSE_proseElement":215,"schema":492,"data":493},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"And in the second one:","paragraph-OGTtWAZF6",{"__TSPROSE_proseElement":215,"schema":496,"data":497,"storageKey":499,"id":500},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":498,"freeze":219},"(m-n)^2 - m^2 - n^2 = \\cancel{m^2} - 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = -2mn","$$ (m-n)^2 - m^2 - n^2 = \\cancel{m^2} - 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = -2mn $$","blockMath-4Ap7uaTLg",{"__TSPROSE_proseElement":215,"schema":502,"children":503,"id":507},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[504],{"__TSPROSE_proseElement":215,"schema":505,"data":506},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Substitute back and cancel:","paragraph-XvQg916sZ",{"__TSPROSE_proseElement":215,"schema":509,"data":510,"storageKey":512,"id":513},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":511,"freeze":219},"\\frac{2\\cancel{mn}}{\\cancel{mn}} \\cdot \\frac{-2\\cancel{mn}}{\\cancel{mn}} = 2 \\cdot (-2) = \\boxed{-4}","$$ \\frac{2\\cancel{mn}}{\\cancel{mn}} \\cdot \\frac{-2\\cancel{mn}}{\\cancel{mn}} = 2 \\cdot (-2) = \\boxed{-4} $$","blockMath-9vwuJBOgx","intro-examples",{"__TSPROSE_proseElement":215,"schema":516,"children":517,"id":530},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[518,521,527],{"__TSPROSE_proseElement":215,"schema":519,"data":520},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"In all three examples, we kept doing the same routine work over and over -- expanding brackets inside powers. Those expansions look very similar -- squares, coefficients of ",{"__TSPROSE_proseElement":215,"schema":522,"data":523,"storageKey":525,"id":526},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"2","$ 2 $","inlinerMath-z9Yar9waf",{"__TSPROSE_proseElement":215,"schema":528,"data":529},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and so on. Instead of wasting time on the same hand calculations every time, mathematicians studied the most common patterns and wrote them down as formulas. That is where the name comes from -- Special Products. Do not confuse them with anything else.","paragraph-yqjJ82Z1o",{"__TSPROSE_proseElement":215,"schema":532,"data":534,"children":536,"id":548},{"name":533,"type":218,"linkable":215,"__TSPROSE_schema":215},"accent_term",{"title":160,"layout":535},"column",[537],{"__TSPROSE_proseElement":215,"schema":538,"children":540},{"name":539,"type":218,"linkable":219,"__TSPROSE_schema":215},"accentMain_term",[541],{"__TSPROSE_proseElement":215,"schema":542,"children":543,"id":547},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[544],{"__TSPROSE_proseElement":215,"schema":545,"data":546},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Formulas that let you quickly \"unpack\" compact powered expressions into expansions or, going the other way, \"pack\" long sums into a compact form. These formulas save you from doing routine calculations by hand.","paragraph-6CjNYA53K","what-are-special-products",{"__TSPROSE_proseElement":215,"schema":550,"data":551,"id":553},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":173,"title":552},"Square of a Sum","square-of-a-sum",{"__TSPROSE_proseElement":215,"schema":555,"children":556,"id":569},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[557,560,566],{"__TSPROSE_proseElement":215,"schema":558,"data":559},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Start with a classic mistake made by ",{"__TSPROSE_proseElement":215,"schema":561,"data":562,"storageKey":564,"id":565},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":563},"90%","$ 90% $","inlinerMath-TriCVV86T",{"__TSPROSE_proseElement":215,"schema":567,"data":568},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," of people, most often school students who are \"not that good\" at math. It is so common that there is even a meme about it:","paragraph-mUzJOMHL6",{"__TSPROSE_proseElement":215,"schema":571,"data":573,"storageKey":574,"id":576},{"name":572,"type":218,"linkable":215,"__TSPROSE_schema":215},"image",{"src":574,"width":575},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fa-b-squared-meme.svg","400px","image-xvwEH7CeL",{"__TSPROSE_proseElement":215,"schema":578,"children":579,"id":602},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[580,583,589,592,599],{"__TSPROSE_proseElement":215,"schema":581,"data":582},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"It would be really nice if the last answer worked that way. People want it so badly that this dream of applying a power directly to the terms ",{"__TSPROSE_proseElement":215,"schema":584,"data":585,"storageKey":587,"id":588},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":586},"(a+b)^n = a^n + b^n","$ (a+b)^n = a^n + b^n $","inlinerMath-1TS0ztSM3",{"__TSPROSE_proseElement":215,"schema":590,"data":591},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," even has its own name -- ",{"__TSPROSE_proseElement":215,"schema":593,"data":595,"storageKey":597,"id":598},{"name":594,"type":234,"linkable":215,"__TSPROSE_schema":215},"referenceInliner",{"label":596},"\"Freshman's Dream\"","\u003Clink:external>\u002Fhttps:\u002F\u002Fw.wiki\u002FPjb","referenceInliner-Y2TDQc1D2",{"__TSPROSE_proseElement":215,"schema":600,"data":601},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". You can see very quickly that this dream does not come true by plugging in actual numbers instead of letters:","paragraph-Nz4QLFaCl",{"__TSPROSE_proseElement":215,"schema":604,"data":605,"storageKey":607,"id":608},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":606,"freeze":219},"\\red{(1+2)^2 = 1^2 + 2^2 = 5} \\\\ \\boxed{\\green{(1 + 2)^2 = 3^2 = 9}} >>{big} \\red{(2+3)^3 = 2^3 + 3^3 = 35} \\\\ \\boxed{\\green{(2 + 3)^3 = 5^3 = 125}}","$$ \\red{(1+2)^2 = 1^2 + 2^2 = 5} \\\\ \\boxed{\\green{(1 + 2)^2 = 3^2 = 9}} >>{big} \\red{(2+3)^3 = 2^3 + 3^3 = 35} \\\\ \\boxed{\\green{(2 + 3)^3 = 5^3 = 125}} $$","blockMath-OQXfuW9pi",{"__TSPROSE_proseElement":215,"schema":610,"children":611,"id":665},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[612,615,621,624,634,637,643,646,655,658,662],{"__TSPROSE_proseElement":215,"schema":613,"data":614},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"In the meme and the first example, the power is two, and the expression ",{"__TSPROSE_proseElement":215,"schema":616,"data":617,"storageKey":619,"id":620},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":618},"(a+b)^2","$ (a+b)^2 $","inlinerMath-K2ZDiwXLY",{"__TSPROSE_proseElement":215,"schema":622,"data":623},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," is called the ",{"__TSPROSE_proseElement":215,"schema":625,"data":627,"children":629,"id":633},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},"emphasis",{"type":628},"bold",[630],{"__TSPROSE_proseElement":215,"schema":631,"data":632},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"\"square of a sum\"","emphasis-ZVmoLZkWb",{"__TSPROSE_proseElement":215,"schema":635,"data":636},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," because the whole sum ",{"__TSPROSE_proseElement":215,"schema":638,"data":639,"storageKey":641,"id":642},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":640},"a + b","$ a + b $","inlinerMath-ZpaGPDgeL",{"__TSPROSE_proseElement":215,"schema":644,"data":645},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," is being squared. That is exactly where the name comes from -- the square of the ",{"__TSPROSE_proseElement":215,"schema":647,"data":648,"children":650,"id":654},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":649},"italic",[651],{"__TSPROSE_proseElement":215,"schema":652,"data":653},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"entire","emphasis-CcnXKWJfa",{"__TSPROSE_proseElement":215,"schema":656,"data":657},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," sum. The correct formula for ",{"__TSPROSE_proseElement":215,"schema":659,"data":660,"storageKey":619,"id":661},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":618},"inlinerMath-K2ZDiwXLY-1",{"__TSPROSE_proseElement":215,"schema":663,"data":664},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," looks a lot like \"Freshman's Dream\", but it is slightly more complicated. You can derive it by hand very quickly. The most direct way is to write the square as a product of two identical binomials and expand the brackets with FOIL:","paragraph-BNwS3zSkj",{"__TSPROSE_proseElement":215,"schema":667,"data":668,"storageKey":669,"children":672,"id":680},{"name":572,"type":218,"linkable":215,"__TSPROSE_schema":215},{"src":669,"width":670,"invert":671},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Ffoil.svg","600px","dark",[673],{"__TSPROSE_proseElement":215,"schema":674,"children":676},{"name":675,"type":234,"linkable":219,"__TSPROSE_schema":215},"caption",[677],{"__TSPROSE_proseElement":215,"schema":678,"data":679},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Multiplying binomials with FOIL","image-SAVXQKdnW",{"__TSPROSE_proseElement":215,"schema":682,"data":683,"storageKey":685,"id":686},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":684,"freeze":219},"(a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \\boxed{a^2 + 2ab + b^2}","$$ (a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \\boxed{a^2 + 2ab + b^2} $$","blockMath-7lei3dGoV",{"__TSPROSE_proseElement":215,"schema":688,"children":689,"id":702},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[690,693,699],{"__TSPROSE_proseElement":215,"schema":691,"data":692},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Going the other way, from the \"sum\" back into the packaged \"product\", is just as simple -- split ",{"__TSPROSE_proseElement":215,"schema":694,"data":695,"storageKey":697,"id":698},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":696},"2ab","$ 2ab $","inlinerMath-McBVEYBhA",{"__TSPROSE_proseElement":215,"schema":700,"data":701},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," into two terms and factor out a common binomial several times in a row:","paragraph-dryCVd4KB",{"__TSPROSE_proseElement":215,"schema":704,"data":705,"storageKey":707,"id":708},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":706,"freeze":219},"a^2 + 2ab + b^2 = a^2 + ab + ab + b^2 = a(a + b) + b(a + b) = (a + b)(a + b) = \\boxed{(a + b)^2}","$$ a^2 + 2ab + b^2 = a^2 + ab + ab + b^2 = a(a + b) + b(a + b) = (a + b)(a + b) = \\boxed{(a + b)^2} $$","blockMath-KeTVimB1e",{"__TSPROSE_proseElement":215,"schema":710,"children":711,"id":759},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[712,715,721,723,729,732,738,741,747,750,756],{"__TSPROSE_proseElement":215,"schema":713,"data":714},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The second derivation is geometric. You can think of ",{"__TSPROSE_proseElement":215,"schema":716,"data":717,"storageKey":719,"id":720},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"a","$ a $","inlinerMath-YqDfaaRXA",{"__TSPROSE_proseElement":215,"schema":722,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":724,"data":725,"storageKey":727,"id":728},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"b","$ b $","inlinerMath-LTS8nNQCZ",{"__TSPROSE_proseElement":215,"schema":730,"data":731},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," as two line segments. Their sum, squared, equals the area of a square whose side is made from those two segments. You can find the total area by adding the areas of the component shapes: the square of area ",{"__TSPROSE_proseElement":215,"schema":733,"data":734,"storageKey":736,"id":737},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":735},"a^2","$ a^2 $","inlinerMath-eiXlIGTfO",{"__TSPROSE_proseElement":215,"schema":739,"data":740},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", two rectangles of area ",{"__TSPROSE_proseElement":215,"schema":742,"data":743,"storageKey":745,"id":746},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":744},"ab","$ ab $","inlinerMath-OAz3gMZNU",{"__TSPROSE_proseElement":215,"schema":748,"data":749},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and the square of area ",{"__TSPROSE_proseElement":215,"schema":751,"data":752,"storageKey":754,"id":755},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":753},"b^2","$ b^2 $","inlinerMath-tBkcHvoEV",{"__TSPROSE_proseElement":215,"schema":757,"data":758},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},".","paragraph-c07zgy9dg",{"__TSPROSE_proseElement":215,"schema":761,"data":762,"storageKey":763,"children":764,"id":771},{"name":572,"type":218,"linkable":215,"__TSPROSE_schema":215},{"src":763,"invert":671},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-sum-schema.svg",[765],{"__TSPROSE_proseElement":215,"schema":766,"children":767},{"name":675,"type":234,"linkable":219,"__TSPROSE_schema":215},[768],{"__TSPROSE_proseElement":215,"schema":769,"data":770},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Geometric derivation of the square of a sum formula","image-Px4gNJeGZ",{"__TSPROSE_proseElement":215,"schema":773,"data":774,"children":775,"id":786},{"name":189,"type":218,"linkable":215,"__TSPROSE_schema":215},{"title":187,"layout":535},[776],{"__TSPROSE_proseElement":215,"schema":777,"children":779},{"name":778,"type":218,"linkable":219,"__TSPROSE_schema":215},"accentMain_statement",[780],{"__TSPROSE_proseElement":215,"schema":781,"data":782,"storageKey":784,"id":785},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":783,"freeze":219},"(a + b)^2 = a^2 + 2ab + b^2","$$ (a + b)^2 = a^2 + 2ab + b^2 $$","blockMath-ntCGb3wGn","square-sum",{"__TSPROSE_proseElement":215,"schema":788,"children":789,"id":804},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[790,793,801],{"__TSPROSE_proseElement":215,"schema":791,"data":792},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"From this point on, you never need to multiply the same two brackets by hand again. It is enough to find three pieces ",{"__TSPROSE_proseElement":215,"schema":794,"data":795,"children":796,"id":800},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[797],{"__TSPROSE_proseElement":215,"schema":798,"data":799},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"from left to right","emphasis-c8VxKaz52",{"__TSPROSE_proseElement":215,"schema":802,"data":803},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", either mentally or on paper: the square of the first term, twice the product of the two terms, and the square of the second term. Then just write them with plus signs. 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With numbers this simple, you should do it mentally:","paragraph-T5YA9lMGp",{"__TSPROSE_proseElement":215,"schema":889,"data":890,"storageKey":892,"id":893},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":891,"freeze":219},"m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25","$$ m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25 $$","blockMath-0KxHw2nNu",{"__TSPROSE_proseElement":215,"schema":895,"children":896,"id":900},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[897],{"__TSPROSE_proseElement":215,"schema":898,"data":899},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Write those pieces with plus signs and the expansion is done:","paragraph-MaZtcGLZc",{"__TSPROSE_proseElement":215,"schema":902,"data":903,"storageKey":905,"id":906},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":904,"freeze":219},"(m+5)^2 = m^2 + 10m + 25","$$ (m+5)^2 = m^2 + 10m + 25 $$","blockMath-6e3NAKPlk",{"__TSPROSE_proseElement":215,"schema":908,"data":909,"children":910},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[911,920,926,934],{"__TSPROSE_proseElement":215,"schema":912,"children":913},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[914],{"__TSPROSE_proseElement":215,"schema":915,"data":916,"storageKey":918,"id":919},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":917,"freeze":219},"\\left( 2 + \\frac{1}{8}x \\right)^2","$$ \\left( 2 + \\frac{1}{8}x \\right)^2 $$","blockMath-f7I6jVg0J",{"__TSPROSE_proseElement":215,"schema":921,"data":922},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":923},{"__ERUDIT_CHECK":215,"name":835,"data":924},{"expr":925},"4 + \\frac{1}{2}x + \\frac{1}{64}x^2",{"__TSPROSE_proseElement":215,"schema":927,"children":928},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[929],{"__TSPROSE_proseElement":215,"schema":930,"data":931,"storageKey":932,"id":933},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":925,"freeze":219},"$$ 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$","blockMath-L506HvSs3",{"__TSPROSE_proseElement":215,"schema":935,"children":936},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[937,970,976,982],{"__TSPROSE_proseElement":215,"schema":938,"children":939,"id":969},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[940,942,946,948,952,954,958,960,966],{"__TSPROSE_proseElement":215,"schema":941,"data":854},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":943,"data":944,"storageKey":719,"id":945},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-2",{"__TSPROSE_proseElement":215,"schema":947,"data":861},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":949,"data":950,"storageKey":525,"id":951},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"inlinerMath-z9Yar9waf-1",{"__TSPROSE_proseElement":215,"schema":953,"data":870},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":955,"data":956,"storageKey":727,"id":957},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-2",{"__TSPROSE_proseElement":215,"schema":959,"data":877},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":961,"data":962,"storageKey":964,"id":965},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":963},"\\frac{1}{8}x","$ \\frac{1}{8}x $","inlinerMath-EYpHaQ5VW",{"__TSPROSE_proseElement":215,"schema":967,"data":968},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". 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In the square of a sum formula, the role of ",{"__TSPROSE_proseElement":215,"schema":1121,"data":1122,"storageKey":719,"id":1123},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-3",{"__TSPROSE_proseElement":215,"schema":1125,"data":861},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1127,"data":1128,"storageKey":1074,"id":1129},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1073},"inlinerMath-80LHk8XHo-2",{"__TSPROSE_proseElement":215,"schema":1131,"data":870},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1133,"data":1134,"storageKey":727,"id":1135},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-3",{"__TSPROSE_proseElement":215,"schema":1137,"data":877},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1139,"data":1140,"storageKey":1104,"id":1141},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1103},"inlinerMath-Djaufz4Sl-1",{"__TSPROSE_proseElement":215,"schema":1143,"data":1144},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". From left to right, find the square of the first term, twice the product of the first and second, and the square of the second.","paragraph-EWbzaIRvs",{"__TSPROSE_proseElement":215,"schema":1147,"data":1148,"storageKey":1150,"id":1151},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1149,"freeze":219},"(-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2","$$ (-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2 $$","blockMath-r7gELRwOl",{"__TSPROSE_proseElement":215,"schema":1153,"children":1154,"id":1157},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1155],{"__TSPROSE_proseElement":215,"schema":1156,"data":899},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-MaZtcGLZc-2",{"__TSPROSE_proseElement":215,"schema":1159,"data":1160,"storageKey":1162,"id":1163},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1161,"freeze":219},"(-7a - 3b)^2 = 49a^2 + 42ab + 9b^2","$$ (-7a - 3b)^2 = 49a^2 + 42ab + 9b^2 $$","blockMath-A2ZrMc5jo",{"__TSPROSE_proseElement":215,"schema":1165,"data":1166,"children":1167},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},"Factoring out the minus sign",[1168,1182,1188,1195],{"__TSPROSE_proseElement":215,"schema":1169,"children":1170,"id":1181},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1171,1174,1178],{"__TSPROSE_proseElement":215,"schema":1172,"data":1173},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Factor a minus sign out of ",{"__TSPROSE_proseElement":215,"schema":1175,"data":1176,"storageKey":1065,"id":1177},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1064},"inlinerMath-EOtl20nRc-1",{"__TSPROSE_proseElement":215,"schema":1179,"data":1180},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},":","paragraph-YuVHUWFZo",{"__TSPROSE_proseElement":215,"schema":1183,"data":1184,"storageKey":1186,"id":1187},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1185,"freeze":219},"-7a - 3b = -(7a + 3b)","$$ -7a - 3b = -(7a + 3b) $$","blockMath-HfGK3huoD",{"__TSPROSE_proseElement":215,"schema":1189,"children":1190,"id":1194},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1191],{"__TSPROSE_proseElement":215,"schema":1192,"data":1193},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Now we have a sum inside the brackets, and the standalone minus sign disappears when squared:","paragraph-9PQSr3iSt",{"__TSPROSE_proseElement":215,"schema":1196,"data":1197,"storageKey":1199,"id":1200},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1198,"freeze":219},"(-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2","$$ (-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2 $$","blockMath-WkRLMEjNE","square-sum-expand-examples",{"__TSPROSE_proseElement":215,"schema":1203,"children":1204,"id":1208},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1205],{"__TSPROSE_proseElement":215,"schema":1206,"data":1207},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The square of a sum formula does not just help you expand brackets quickly. It also lets you go the other way -- pack an expanded expression back into a bracket squared, into the square of a sum. This process is often called completing the square, and we will talk about it separately below.","paragraph-GYzlgIrrc",{"__TSPROSE_proseElement":215,"schema":1210,"children":1211,"id":1266},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1212,1215,1219,1221,1225,1228,1232,1235,1243,1246,1250,1252,1256,1259,1263],{"__TSPROSE_proseElement":215,"schema":1213,"data":1214},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Packing is slightly harder than expanding. The main goal is to find ",{"__TSPROSE_proseElement":215,"schema":1216,"data":1217,"storageKey":719,"id":1218},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-4",{"__TSPROSE_proseElement":215,"schema":1220,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1222,"data":1223,"storageKey":727,"id":1224},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-4",{"__TSPROSE_proseElement":215,"schema":1226,"data":1227},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," so you can build the square of a sum ",{"__TSPROSE_proseElement":215,"schema":1229,"data":1230,"storageKey":619,"id":1231},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":618},"inlinerMath-K2ZDiwXLY-2",{"__TSPROSE_proseElement":215,"schema":1233,"data":1234},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". There are two ways to find them. The first and fastest way is to look at the ",{"__TSPROSE_proseElement":215,"schema":1236,"data":1237,"children":1238,"id":1242},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[1239],{"__TSPROSE_proseElement":215,"schema":1240,"data":1241},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"outer terms","emphasis-yerzy2RVi",{"__TSPROSE_proseElement":215,"schema":1244,"data":1245},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},": in simple cases, you immediately see perfect squares there, which instantly reveal ",{"__TSPROSE_proseElement":215,"schema":1247,"data":1248,"storageKey":719,"id":1249},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-5",{"__TSPROSE_proseElement":215,"schema":1251,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1253,"data":1254,"storageKey":727,"id":1255},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-5",{"__TSPROSE_proseElement":215,"schema":1257,"data":1258},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". After that, you only need to check that ",{"__TSPROSE_proseElement":215,"schema":1260,"data":1261,"storageKey":697,"id":1262},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":696},"inlinerMath-McBVEYBhA-1",{"__TSPROSE_proseElement":215,"schema":1264,"data":1265},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," matches the middle term, and the packing is done.","paragraph-7v4ST7YYR",{"__TSPROSE_proseElement":215,"schema":1268,"children":1269,"id":1324},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1270,1273,1281,1284,1288,1290,1294,1297,1301,1304,1308,1311,1315,1318,1322],{"__TSPROSE_proseElement":215,"schema":1271,"data":1272},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The second way is to start with the ",{"__TSPROSE_proseElement":215,"schema":1274,"data":1275,"children":1276,"id":1280},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[1277],{"__TSPROSE_proseElement":215,"schema":1278,"data":1279},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"middle term","emphasis-NhuCl9oix",{"__TSPROSE_proseElement":215,"schema":1282,"data":1283},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},": it contains all the information about ",{"__TSPROSE_proseElement":215,"schema":1285,"data":1286,"storageKey":719,"id":1287},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-6",{"__TSPROSE_proseElement":215,"schema":1289,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1291,"data":1292,"storageKey":727,"id":1293},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-6",{"__TSPROSE_proseElement":215,"schema":1295,"data":1296},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Take the middle term, divide it by ",{"__TSPROSE_proseElement":215,"schema":1298,"data":1299,"storageKey":525,"id":1300},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"inlinerMath-z9Yar9waf-2",{"__TSPROSE_proseElement":215,"schema":1302,"data":1303},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and you get the product ",{"__TSPROSE_proseElement":215,"schema":1305,"data":1306,"storageKey":745,"id":1307},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":744},"inlinerMath-OAz3gMZNU-1",{"__TSPROSE_proseElement":215,"schema":1309,"data":1310},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Then you only need to figure out which part is ",{"__TSPROSE_proseElement":215,"schema":1312,"data":1313,"storageKey":719,"id":1314},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-7",{"__TSPROSE_proseElement":215,"schema":1316,"data":1317},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," and which part is ",{"__TSPROSE_proseElement":215,"schema":1319,"data":1320,"storageKey":727,"id":1321},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-7",{"__TSPROSE_proseElement":215,"schema":1323,"data":758},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-dlQ81ZW8x",{"__TSPROSE_proseElement":215,"schema":1326,"data":1327,"children":1330,"id":2062},{"name":239,"type":218,"linkable":215,"__TSPROSE_schema":215},{"title":1328,"level":242,"attributes":1329},"Packing into the square of a sum",[],[1331,1338,1506,1691,1867],{"__TSPROSE_proseElement":215,"schema":1332,"children":1333,"id":1337},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1334],{"__TSPROSE_proseElement":215,"schema":1335,"data":1336},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Write the sum as a square of a sum:","paragraph-Yyr9gOxTC",{"__TSPROSE_proseElement":215,"schema":1339,"data":1340,"children":1341},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[1342,1351,1357,1365],{"__TSPROSE_proseElement":215,"schema":1343,"children":1344},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[1345],{"__TSPROSE_proseElement":215,"schema":1346,"data":1347,"storageKey":1349,"id":1350},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1348,"freeze":219},"49 + 14x + x^2","$$ 49 + 14x + x^2 $$","blockMath-lLCEZPE65",{"__TSPROSE_proseElement":215,"schema":1352,"data":1353},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":1354},{"__ERUDIT_CHECK":215,"name":835,"data":1355},{"expr":1356},"(7 + x)^2",{"__TSPROSE_proseElement":215,"schema":1358,"children":1359},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[1360],{"__TSPROSE_proseElement":215,"schema":1361,"data":1362,"storageKey":1363,"id":1364},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1356,"freeze":219},"$$ (7 + x)^2 $$","blockMath-k4z5iytxW",{"__TSPROSE_proseElement":215,"schema":1366,"children":1367},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[1368,1428],{"__TSPROSE_proseElement":215,"schema":1369,"data":1370,"children":1371},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},"Using the outer terms",[1372,1422],{"__TSPROSE_proseElement":215,"schema":1373,"children":1374,"id":1421},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1375,1378,1384,1386,1392,1395,1401,1403,1409,1412,1418],{"__TSPROSE_proseElement":215,"schema":1376,"data":1377},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Look at the outer terms: ",{"__TSPROSE_proseElement":215,"schema":1379,"data":1380,"storageKey":1382,"id":1383},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1381},"49 = 7^2","$ 49 = 7^2 $","inlinerMath-ocltGy9bI",{"__TSPROSE_proseElement":215,"schema":1385,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1387,"data":1388,"storageKey":1390,"id":1391},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1389},"x^2 = x^2","$ x^2 = x^2 $","inlinerMath-swnzwUffe",{"__TSPROSE_proseElement":215,"schema":1393,"data":1394},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Both are nice perfect squares, so you can immediately see ",{"__TSPROSE_proseElement":215,"schema":1396,"data":1397,"storageKey":1399,"id":1400},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1398},"a = 7","$ a = 7 $","inlinerMath-CiXrICT5m",{"__TSPROSE_proseElement":215,"schema":1402,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1404,"data":1405,"storageKey":1407,"id":1408},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1406},"b = x","$ b = x $","inlinerMath-wIHP4wnz6",{"__TSPROSE_proseElement":215,"schema":1410,"data":1411},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Check the middle term: ",{"__TSPROSE_proseElement":215,"schema":1413,"data":1414,"storageKey":1416,"id":1417},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1415},"2ab = 2 \\cdot 7 \\cdot x = 14x","$ 2ab = 2 \\cdot 7 \\cdot x = 14x $","inlinerMath-aETNmkPHY",{"__TSPROSE_proseElement":215,"schema":1419,"data":1420},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"-- it matches. Write the result:","paragraph-TZcMThCwB",{"__TSPROSE_proseElement":215,"schema":1423,"data":1424,"storageKey":1426,"id":1427},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1425,"freeze":219},"49 + 14x + x^2 = \\underset{a^2}{7^2} + 2 \\cdot \\underset{a}{7} \\cdot \\underset{b}{x} + \\underset{b^2}{x^2} = (7 + x)^2","$$ 49 + 14x + x^2 = \\underset{a^2}{7^2} + 2 \\cdot \\underset{a}{7} \\cdot \\underset{b}{x} + \\underset{b^2}{x^2} = (7 + x)^2 $$","blockMath-eSZvu6RZE",{"__TSPROSE_proseElement":215,"schema":1429,"data":1430,"children":1431},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},"Using the middle term",[1432,1502],{"__TSPROSE_proseElement":215,"schema":1433,"children":1434,"id":1501},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1435,1438,1444,1447,1451,1454,1460,1463,1469,1471,1477,1480,1486,1489,1493,1495,1499],{"__TSPROSE_proseElement":215,"schema":1436,"data":1437},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Start with the middle term ",{"__TSPROSE_proseElement":215,"schema":1439,"data":1440,"storageKey":1442,"id":1443},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1441},"14x","$ 14x $","inlinerMath-wmtD0OrpQ",{"__TSPROSE_proseElement":215,"schema":1445,"data":1446},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Divide it by ",{"__TSPROSE_proseElement":215,"schema":1448,"data":1449,"storageKey":525,"id":1450},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"inlinerMath-z9Yar9waf-3",{"__TSPROSE_proseElement":215,"schema":1452,"data":1453},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," and get ",{"__TSPROSE_proseElement":215,"schema":1455,"data":1456,"storageKey":1458,"id":1459},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1457},"7x","$ 7x $","inlinerMath-ESNBsoN9g",{"__TSPROSE_proseElement":215,"schema":1461,"data":1462},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Now split that product into two factors whose squares match the outer terms ",{"__TSPROSE_proseElement":215,"schema":1464,"data":1465,"storageKey":1467,"id":1468},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1466},"49","$ 49 $","inlinerMath-CKZbQKHFg",{"__TSPROSE_proseElement":215,"schema":1470,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1472,"data":1473,"storageKey":1475,"id":1476},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1474},"x^2","$ x^2 $","inlinerMath-tzIa3VWcL",{"__TSPROSE_proseElement":215,"schema":1478,"data":1479},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". 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Check: ",{"__TSPROSE_proseElement":215,"schema":1670,"data":1671,"storageKey":1673,"id":1674},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1672},"1^2 = 1 = a","$ 1^2 = 1 = a $","inlinerMath-UKP1QET9S",{"__TSPROSE_proseElement":215,"schema":1676,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1678,"data":1679,"storageKey":1681,"id":1682},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1680},"(4y)^2 = 16y^2 = b^2","$ (4y)^2 = 16y^2 = b^2 $","inlinerMath-RPKOx7uC1",{"__TSPROSE_proseElement":215,"schema":1684,"data":1685},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Everything matches. 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Do not let that stop you...","paragraph-YahlCIZRo",{"__TSPROSE_proseElement":215,"schema":1721,"children":1722},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[1723],{"__TSPROSE_proseElement":215,"schema":1724,"data":1725,"storageKey":1727,"id":1728},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1726,"freeze":219},"\\left(\\frac{k}{2} + 1 \\right)^2","$$ \\left(\\frac{k}{2} + 1 \\right)^2 $$","blockMath-Sqh9IOrVG",{"__TSPROSE_proseElement":215,"schema":1730,"children":1731},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[1732,1786],{"__TSPROSE_proseElement":215,"schema":1733,"data":1370,"children":1734},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},[1735,1780],{"__TSPROSE_proseElement":215,"schema":1736,"children":1737,"id":1779},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1738,1740,1746,1748,1752,1754,1760,1762,1768,1770,1776],{"__TSPROSE_proseElement":215,"schema":1739,"data":1377},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1741,"data":1742,"storageKey":1744,"id":1745},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1743},"\\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2","$ \\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2 $","inlinerMath-TfEAXhY2F",{"__TSPROSE_proseElement":215,"schema":1747,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1749,"data":1750,"storageKey":1564,"id":1751},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1563},"inlinerMath-AM9HQ4pvg-1",{"__TSPROSE_proseElement":215,"schema":1753,"data":1394},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1755,"data":1756,"storageKey":1758,"id":1759},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1757},"a = \\frac{k}{2}","$ a = \\frac{k}{2} $","inlinerMath-EqVNcAV7k",{"__TSPROSE_proseElement":215,"schema":1761,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1763,"data":1764,"storageKey":1766,"id":1767},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1765},"b = 1","$ b = 1 $","inlinerMath-jT8WRtcC5",{"__TSPROSE_proseElement":215,"schema":1769,"data":1411},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1771,"data":1772,"storageKey":1774,"id":1775},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1773},"2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k","$ 2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k $","inlinerMath-h10U39IqB",{"__TSPROSE_proseElement":215,"schema":1777,"data":1778},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," -- it matches. 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That gives ",{"__TSPROSE_proseElement":215,"schema":1810,"data":1811,"storageKey":1813,"id":1814},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1812},"\\frac{k}{2}","$ \\frac{k}{2} $","inlinerMath-IcB1YEh3y",{"__TSPROSE_proseElement":215,"schema":1816,"data":1817},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Now write that fraction as a product of two factors whose squares match the outer terms ",{"__TSPROSE_proseElement":215,"schema":1819,"data":1820,"storageKey":1822,"id":1823},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1821},"\\frac{1}{4}k^2","$ \\frac{1}{4}k^2 $","inlinerMath-XhsYjlucP",{"__TSPROSE_proseElement":215,"schema":1825,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1827,"data":1828,"storageKey":1020,"id":1829},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":385},"inlinerMath-GWJ3eIxvK-2",{"__TSPROSE_proseElement":215,"schema":1831,"data":1652},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1833,"data":1834,"storageKey":1538,"id":1835},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1537},"inlinerMath-ooHpGun5c-2",{"__TSPROSE_proseElement":215,"schema":1837,"data":1659},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1839,"data":1840,"storageKey":1842,"id":1843},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1841},"\\frac{k}{2} \\cdot 1","$ \\frac{k}{2} \\cdot 1 $","inlinerMath-5VcE9HQud",{"__TSPROSE_proseElement":215,"schema":1845,"data":1668},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1847,"data":1848,"storageKey":1850,"id":1851},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1849},"\\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2","$ \\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2 $","inlinerMath-9J027juAN",{"__TSPROSE_proseElement":215,"schema":1853,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1855,"data":1856,"storageKey":1858,"id":1859},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1857},"1^2 = 1 = b^2","$ 1^2 = 1 = b^2 $","inlinerMath-5XaGNX6Ix",{"__TSPROSE_proseElement":215,"schema":1861,"data":1685},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-N5QsfrYkL",{"__TSPROSE_proseElement":215,"schema":1864,"data":1865,"storageKey":1784,"id":1866},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1783,"freeze":219},"blockMath-OAtFOiXrE-1",{"__TSPROSE_proseElement":215,"schema":1868,"data":1869,"children":1870},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[1871,1880,1886,1896,1904],{"__TSPROSE_proseElement":215,"schema":1872,"children":1873},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[1874],{"__TSPROSE_proseElement":215,"schema":1875,"data":1876,"storageKey":1878,"id":1879},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1877,"freeze":219},"16t^2 + 36m^2 + 48tm","$$ 16t^2 + 36m^2 + 48tm $$","blockMath-BRO83bY8l",{"__TSPROSE_proseElement":215,"schema":1881,"data":1882},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":1883},{"__ERUDIT_CHECK":215,"name":835,"data":1884},{"expr":1885},"(4t + 6m)^2",{"__TSPROSE_proseElement":215,"schema":1887,"children":1888},{"name":1009,"type":218,"linkable":219,"__TSPROSE_schema":215},[1889],{"__TSPROSE_proseElement":215,"schema":1890,"children":1891,"id":1895},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1892],{"__TSPROSE_proseElement":215,"schema":1893,"data":1894},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The terms in the sum do not always appear in the same order as in the square of a sum formula. In cases like this, you can reorder them, because swapping terms does not change a sum.","paragraph-vreulrYsX",{"__TSPROSE_proseElement":215,"schema":1897,"children":1898},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[1899],{"__TSPROSE_proseElement":215,"schema":1900,"data":1901,"storageKey":1902,"id":1903},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1885,"freeze":219},"$$ (4t + 6m)^2 $$","blockMath-YexRM6ll2",{"__TSPROSE_proseElement":215,"schema":1905,"children":1906},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[1907,1914,1920,1975],{"__TSPROSE_proseElement":215,"schema":1908,"children":1909,"id":1913},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[1910],{"__TSPROSE_proseElement":215,"schema":1911,"data":1912},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"This sum looks unusual: the second term is also a square. Swap the second and third terms (the sum does not change) so the pure squares sit on the outside and the mixed term is in the middle:","paragraph-UAv9wOl7q",{"__TSPROSE_proseElement":215,"schema":1915,"data":1916,"storageKey":1918,"id":1919},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1917,"freeze":219},"16t^2 + 48tm + 36m^2","$$ 16t^2 + 48tm + 36m^2 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(6m)^2","$ 36m^2 = (6m)^2 $","inlinerMath-44R58sl9K",{"__TSPROSE_proseElement":215,"schema":1943,"data":1394},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1945,"data":1946,"storageKey":1948,"id":1949},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1947},"a = 4t","$ a = 4t $","inlinerMath-Dvo0ZB2yR",{"__TSPROSE_proseElement":215,"schema":1951,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1953,"data":1954,"storageKey":1956,"id":1957},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1955},"b = 6m","$ b = 6m $","inlinerMath-zfqINXAo1",{"__TSPROSE_proseElement":215,"schema":1959,"data":1411},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":1961,"data":1962,"storageKey":1964,"id":1965},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1963},"2ab = 2 \\cdot 4t \\cdot 6m = 48tm","$ 2ab = 2 \\cdot 4t \\cdot 6m = 48tm $","inlinerMath-ukNlMgNXk",{"__TSPROSE_proseElement":215,"schema":1967,"data":1778},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-EANb24h54",{"__TSPROSE_proseElement":215,"schema":1970,"data":1971,"storageKey":1973,"id":1974},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":1972,"freeze":219},"16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2","$$ 16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2 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Euclid, for example, was already using them geometrically in the 3rd century BCE to compute areas. He stated the square of a sum formula like this:","paragraph-2Y370wJJI",{"__TSPROSE_proseElement":215,"schema":2078,"data":2079,"children":2080,"id":2089},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},{"center":215},[2081],{"__TSPROSE_proseElement":215,"schema":2082,"data":2083,"children":2084,"id":2088},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":649},[2085],{"__TSPROSE_proseElement":215,"schema":2086,"data":2087},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"\"If a segment is divided into two parts in any way, then the area of the square built on the whole segment is equal to the sum of the areas of the squares built on each part, plus twice the area of the rectangle whose sides are those two parts.\"","emphasis-xlCaZZu23","paragraph-I83pooTxi",{"__TSPROSE_proseElement":215,"schema":2091,"children":2092,"id":2105},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2093,2096,2102],{"__TSPROSE_proseElement":215,"schema":2094,"data":2095},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Special Products took on their modern form much later, in the 16th and 17th centuries, thanks to mathematicians Francois Viete (the same Viete whose name appears in ",{"__TSPROSE_proseElement":215,"schema":2097,"data":2098,"storageKey":2100,"id":2101},{"name":594,"type":234,"linkable":215,"__TSPROSE_schema":215},{"label":2099},"Vieta's formulas","\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas","referenceInliner-NZYwv1qIT",{"__TSPROSE_proseElement":215,"schema":2103,"data":2104},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},") and Rene Descartes. These formulas are still some of the most frequently used mathematical tricks everywhere in mathematics, from simplifying expressions in algebra to factoring equations in cryptography.","paragraph-e4fz6Gbji","callout-XNMgnrSoJ",{"__TSPROSE_proseElement":215,"schema":2108,"data":2109,"id":2111},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":173,"title":2110},"Square of a Difference","square-of-a-difference",{"__TSPROSE_proseElement":215,"schema":2113,"children":2114,"id":2146},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2115,2118,2124,2127,2133,2136,2143],{"__TSPROSE_proseElement":215,"schema":2116,"data":2117},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"By analogy with the square of a sum, an expression of the form ",{"__TSPROSE_proseElement":215,"schema":2119,"data":2120,"storageKey":2122,"id":2123},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2121},"(a-b)^2","$ (a-b)^2 $","inlinerMath-kgXTO8hrU",{"__TSPROSE_proseElement":215,"schema":2125,"data":2126},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," is called the square of a difference. The reason is clear: we have a difference of two numbers ",{"__TSPROSE_proseElement":215,"schema":2128,"data":2129,"storageKey":2131,"id":2132},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2130},"a-b","$ a-b $","inlinerMath-zxME95g2R",{"__TSPROSE_proseElement":215,"schema":2134,"data":2135},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and we want to square that entire difference, that is, raise it to the second power. That is where the name comes from: the square of the ",{"__TSPROSE_proseElement":215,"schema":2137,"data":2138,"children":2139,"id":2142},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":649},[2140],{"__TSPROSE_proseElement":215,"schema":2141,"data":653},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"emphasis-CcnXKWJfa-1",{"__TSPROSE_proseElement":215,"schema":2144,"data":2145},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," difference.","paragraph-mkBXm8rvQ",{"__TSPROSE_proseElement":215,"schema":2148,"children":2149,"id":2153},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2150],{"__TSPROSE_proseElement":215,"schema":2151,"data":2152},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Algebraically, the square-of-a-difference formula is derived in both directions exactly the same way as the square of a sum: either expand with FOIL in one direction, or split the doubled term and factor out common factors in the other:","paragraph-Ie3Ba49ny",{"__TSPROSE_proseElement":215,"schema":2155,"data":2156,"storageKey":2158,"id":2159},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2157,"freeze":219},"(a - b)^2 = (a - b)(a - b) = a^2 - ab - ba + b^2 = \\boxed{a^2 - 2ab + b^2} \\\\ a^2 - 2ab + b^2 = a^2 - ab - ab + b^2 = a(a - b) - b(a - b) = (a - b)(a - b) = \\boxed{(a - b)^2}","$$ (a - b)^2 = (a - b)(a - b) = a^2 - ab - ba + b^2 = \\boxed{a^2 - 2ab + b^2} \\\\ a^2 - 2ab + b^2 = a^2 - ab - ab + b^2 = a(a - b) - b(a - b) = (a - b)(a - b) = \\boxed{(a - b)^2} $$","blockMath-4ExN6IPzS",{"__TSPROSE_proseElement":215,"schema":2161,"children":2162,"id":2186},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2163,2166,2170,2173,2177,2180,2184],{"__TSPROSE_proseElement":215,"schema":2164,"data":2165},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"A geometric derivation is possible too. For the square of a sum, we found the total area of a large square whose side was made of two segments. Now we already have a large square with side ",{"__TSPROSE_proseElement":215,"schema":2167,"data":2168,"storageKey":719,"id":2169},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-8",{"__TSPROSE_proseElement":215,"schema":2171,"data":2172},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and we shorten its sides by the length ",{"__TSPROSE_proseElement":215,"schema":2174,"data":2175,"storageKey":727,"id":2176},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-8",{"__TSPROSE_proseElement":215,"schema":2178,"data":2179},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". The area of the smaller square is exactly ",{"__TSPROSE_proseElement":215,"schema":2181,"data":2182,"storageKey":2122,"id":2183},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2121},"inlinerMath-kgXTO8hrU-1",{"__TSPROSE_proseElement":215,"schema":2185,"data":1180},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-hvoOLtMoF",{"__TSPROSE_proseElement":215,"schema":2188,"data":2189,"storageKey":2190,"children":2191,"id":2198},{"name":572,"type":218,"linkable":215,"__TSPROSE_schema":215},{"src":2190,"invert":671},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-diff-schema.svg",[2192],{"__TSPROSE_proseElement":215,"schema":2193,"children":2194},{"name":675,"type":234,"linkable":219,"__TSPROSE_schema":215},[2195],{"__TSPROSE_proseElement":215,"schema":2196,"data":2197},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Geometric derivation of the square-of-a-difference formula","image-CK17SQR3S",{"__TSPROSE_proseElement":215,"schema":2200,"data":2201,"children":2202,"id":2212},{"name":189,"type":218,"linkable":215,"__TSPROSE_schema":215},{"title":192,"layout":535},[2203],{"__TSPROSE_proseElement":215,"schema":2204,"children":2205},{"name":778,"type":218,"linkable":219,"__TSPROSE_schema":215},[2206],{"__TSPROSE_proseElement":215,"schema":2207,"data":2208,"storageKey":2210,"id":2211},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2209,"freeze":219},"(a - b)^2 = a^2 - 2ab + b^2","$$ (a - b)^2 = a^2 - 2ab + b^2 $$","blockMath-xpqkz7Ecb","square-diff",{"__TSPROSE_proseElement":215,"schema":2214,"children":2215,"id":2219},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2216],{"__TSPROSE_proseElement":215,"schema":2217,"data":2218},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"As you can see, the square of a difference formula differs from the square of a sum formula only by changing the first sign from plus to minus. It is enough to remember the plus version and, when needed, change the first sign to minus.","paragraph-BhuO5l0Ch",{"__TSPROSE_proseElement":215,"schema":2221,"data":2223,"children":2225,"id":2290},{"name":2222,"type":218,"linkable":215,"__TSPROSE_schema":215},"accent_important",{"title":2224,"layout":535},"There Is a Minus Sign, but It Is Still a Sum!",[2226],{"__TSPROSE_proseElement":215,"schema":2227,"children":2229},{"name":2228,"type":218,"linkable":219,"__TSPROSE_schema":215},"accentMain_important",[2230,2266,2272],{"__TSPROSE_proseElement":215,"schema":2231,"children":2232,"id":2265},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2233,2236,2244,2247,2253,2256,2262],{"__TSPROSE_proseElement":215,"schema":2234,"data":2235},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Because the minus sign has two roles in mathematics, ",{"__TSPROSE_proseElement":215,"schema":2237,"data":2238,"children":2239,"id":2243},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628,"accent":215},[2240],{"__TSPROSE_proseElement":215,"schema":2241,"data":2242},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"any difference can be written as a sum","emphasis-lDo8vX2Ed",{"__TSPROSE_proseElement":215,"schema":2245,"data":2246},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," if the minus signs are understood as negation. Not three minus two ",{"__TSPROSE_proseElement":215,"schema":2248,"data":2249,"storageKey":2251,"id":2252},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2250},"3-2","$ 3-2 $","inlinerMath-D102lV2Oh",{"__TSPROSE_proseElement":215,"schema":2254,"data":2255},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", but three plus negative two ",{"__TSPROSE_proseElement":215,"schema":2257,"data":2258,"storageKey":2260,"id":2261},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2259},"3+(-2)","$ 3+(-2) $","inlinerMath-rXqEDLuYB",{"__TSPROSE_proseElement":215,"schema":2263,"data":2264},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". This works for expressions too:","paragraph-Fx2IdBEwr",{"__TSPROSE_proseElement":215,"schema":2267,"data":2268,"storageKey":2270,"id":2271},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2269,"freeze":219},"- a - b + c - d = (-a) + (-b) + c + (-d)","$$ - a - b + c - d = (-a) + (-b) + c + (-d) $$","blockMath-bBmMzTW6H",{"__TSPROSE_proseElement":215,"schema":2273,"children":2274,"id":2289},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2275,2278,2286],{"__TSPROSE_proseElement":215,"schema":2276,"data":2277},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"So do not be surprised when mathematicians call some expressions sums, ",{"__TSPROSE_proseElement":215,"schema":2279,"data":2280,"children":2281,"id":2285},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[2282],{"__TSPROSE_proseElement":215,"schema":2283,"data":2284},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"even if there are no plus signs at all!","emphasis-gC3DQEdbk",{"__TSPROSE_proseElement":215,"schema":2287,"data":2288},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," We will do the same.","paragraph-0ixJoc7V2","there-is-a-minus-sign-but-it-is-still-a-sum",{"__TSPROSE_proseElement":215,"schema":2292,"children":2293,"id":2313},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2294,2297,2301,2304,2310],{"__TSPROSE_proseElement":215,"schema":2295,"data":2296},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Practice using the square-of-a-difference formula to expand parentheses quickly and to factor expressions back into them. The usage pattern is exactly the same as for the square of a sum; the main thing is not to get lost in the minus signs. When factoring back into parentheses, you divide not by ",{"__TSPROSE_proseElement":215,"schema":2298,"data":2299,"storageKey":525,"id":2300},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"inlinerMath-z9Yar9waf-7",{"__TSPROSE_proseElement":215,"schema":2302,"data":2303},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," but by ",{"__TSPROSE_proseElement":215,"schema":2305,"data":2306,"storageKey":2308,"id":2309},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2307},"-2","$ -2 $","inlinerMath-02bjIi1Ry",{"__TSPROSE_proseElement":215,"schema":2311,"data":2312},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". This time we will combine both exercises, expansion and factoring, into one set:","paragraph-13umLls6J",{"__TSPROSE_proseElement":215,"schema":2315,"data":2316,"children":2319,"id":3249},{"name":239,"type":218,"linkable":215,"__TSPROSE_schema":215},{"title":2317,"level":242,"attributes":2318},"Examples Using the Square of a Difference",[],[2320,2456,2660,2774,2976,3067],{"__TSPROSE_proseElement":215,"schema":2321,"data":2322,"children":2323},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[2324,2340,2346,2354,2415],{"__TSPROSE_proseElement":215,"schema":2325,"children":2326},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[2327,2334],{"__TSPROSE_proseElement":215,"schema":2328,"children":2329,"id":2333},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2330],{"__TSPROSE_proseElement":215,"schema":2331,"data":2332},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Expand the parentheses:","paragraph-qYJVaoVtK",{"__TSPROSE_proseElement":215,"schema":2335,"data":2336,"storageKey":2338,"id":2339},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2337,"freeze":219},"(6-c)^2","$$ (6-c)^2 $$","blockMath-ekLbCsNP4",{"__TSPROSE_proseElement":215,"schema":2341,"data":2342},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":2343},{"__ERUDIT_CHECK":215,"name":835,"data":2344},{"expr":2345},"36 - 12c + c^2",{"__TSPROSE_proseElement":215,"schema":2347,"children":2348},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[2349],{"__TSPROSE_proseElement":215,"schema":2350,"data":2351,"storageKey":2352,"id":2353},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2345,"freeze":219},"$$ 36 - 12c + c^2 $$","blockMath-LDyuZrOUH",{"__TSPROSE_proseElement":215,"schema":2355,"children":2356},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[2357,2396,2402,2409],{"__TSPROSE_proseElement":215,"schema":2358,"children":2359,"id":2395},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2360,2363,2367,2370,2376,2379,2383,2386,2392],{"__TSPROSE_proseElement":215,"schema":2361,"data":2362},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Here ",{"__TSPROSE_proseElement":215,"schema":2364,"data":2365,"storageKey":719,"id":2366},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-9",{"__TSPROSE_proseElement":215,"schema":2368,"data":2369},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," is played by the number ",{"__TSPROSE_proseElement":215,"schema":2371,"data":2372,"storageKey":2374,"id":2375},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2373},"6","$ 6 $","inlinerMath-9rSWhNw25",{"__TSPROSE_proseElement":215,"schema":2377,"data":2378},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and ",{"__TSPROSE_proseElement":215,"schema":2380,"data":2381,"storageKey":727,"id":2382},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-9",{"__TSPROSE_proseElement":215,"schema":2384,"data":2385},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," by the letter ",{"__TSPROSE_proseElement":215,"schema":2387,"data":2388,"storageKey":2390,"id":2391},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2389},"c","$ c $","inlinerMath-8Adf0APnu",{"__TSPROSE_proseElement":215,"schema":2393,"data":2394},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". From left to right, find the square of the first term, the negative of twice the product of the first and second terms, and the square of the second term. With numbers this simple, it is best to do the computation mentally:","paragraph-YaZv649zg",{"__TSPROSE_proseElement":215,"schema":2397,"data":2398,"storageKey":2400,"id":2401},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2399,"freeze":219},"6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2","$$ 6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2 $$","blockMath-goZZUGvb0",{"__TSPROSE_proseElement":215,"schema":2403,"children":2404,"id":2408},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2405],{"__TSPROSE_proseElement":215,"schema":2406,"data":2407},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Write all the results together:","paragraph-xv2qdT6ML",{"__TSPROSE_proseElement":215,"schema":2410,"data":2411,"storageKey":2413,"id":2414},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2412,"freeze":219},"(6-c)^2 = 36 - 12c + c^2","$$ (6-c)^2 = 36 - 12c + c^2 $$","blockMath-1J8WJP6EN",{"__TSPROSE_proseElement":215,"schema":2416,"children":2418},{"name":2417,"type":218,"linkable":219,"__TSPROSE_schema":215},"problemNote",[2419,2435,2441],{"__TSPROSE_proseElement":215,"schema":2420,"children":2421,"id":2434},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2422,2425,2431],{"__TSPROSE_proseElement":215,"schema":2423,"data":2424},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Many beginners often make the double minus mistake. When applying the square-of-a-difference formula, they take one minus sign from the formula and a second one from ",{"__TSPROSE_proseElement":215,"schema":2426,"data":2427,"storageKey":2429,"id":2430},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2428},"-b","$ -b $","inlinerMath-MMRLAwV3s",{"__TSPROSE_proseElement":215,"schema":2432,"data":2433},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". As a result, minus times minus becomes plus, and they get the wrong answer:","paragraph-NxEixv6Kh",{"__TSPROSE_proseElement":215,"schema":2436,"data":2437,"storageKey":2439,"id":2440},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2438,"freeze":219},"(6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2}","$$ (6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2} $$","blockMath-ucfrketpu",{"__TSPROSE_proseElement":215,"schema":2442,"children":2443,"id":2455},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2444,2447],{"__TSPROSE_proseElement":215,"schema":2445,"data":2446},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Remember once and for all: ",{"__TSPROSE_proseElement":215,"schema":2448,"data":2449,"children":2450,"id":2454},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[2451],{"__TSPROSE_proseElement":215,"schema":2452,"data":2453},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"any minus sign has already been accounted for in the square of a difference. Do not think about signs separately at all!","emphasis-scv1WyRre","paragraph-316s7d4GN",{"__TSPROSE_proseElement":215,"schema":2457,"data":2458,"children":2459},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[2460,2476,2482,2499,2507],{"__TSPROSE_proseElement":215,"schema":2461,"children":2462},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[2463,2470],{"__TSPROSE_proseElement":215,"schema":2464,"children":2465,"id":2469},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2466],{"__TSPROSE_proseElement":215,"schema":2467,"data":2468},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Write the sum as a square of a difference:","paragraph-MudZgQrHU",{"__TSPROSE_proseElement":215,"schema":2471,"data":2472,"storageKey":2474,"id":2475},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2473,"freeze":219},"9x^2 - 6x + 1","$$ 9x^2 - 6x + 1 $$","blockMath-1DwYljmox",{"__TSPROSE_proseElement":215,"schema":2477,"data":2478},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":2479},{"__ERUDIT_CHECK":215,"name":835,"data":2480},{"expr":2481},"(3x - 1)^2",{"__TSPROSE_proseElement":215,"schema":2483,"children":2484},{"name":1009,"type":218,"linkable":219,"__TSPROSE_schema":215},[2485],{"__TSPROSE_proseElement":215,"schema":2486,"children":2487,"id":2498},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2488,2491,2495],{"__TSPROSE_proseElement":215,"schema":2489,"data":2490},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"You have not forgotten that ",{"__TSPROSE_proseElement":215,"schema":2492,"data":2493,"storageKey":1538,"id":2494},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1537},"inlinerMath-ooHpGun5c-3",{"__TSPROSE_proseElement":215,"schema":2496,"data":2497},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", have you?","paragraph-Prk67niCV",{"__TSPROSE_proseElement":215,"schema":2500,"children":2501},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[2502],{"__TSPROSE_proseElement":215,"schema":2503,"data":2504,"storageKey":2505,"id":2506},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2481,"freeze":219},"$$ (3x - 1)^2 $$","blockMath-p7CiNdIap",{"__TSPROSE_proseElement":215,"schema":2508,"children":2509},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[2510,2564],{"__TSPROSE_proseElement":215,"schema":2511,"data":2512,"children":2513},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},"Using the Outer Terms",[2514,2558],{"__TSPROSE_proseElement":215,"schema":2515,"children":2516,"id":2557},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2517,2519,2525,2527,2531,2534,2540,2542,2546,2548,2554],{"__TSPROSE_proseElement":215,"schema":2518,"data":1377},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2520,"data":2521,"storageKey":2523,"id":2524},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2522},"9x^2 = (3x)^2","$ 9x^2 = (3x)^2 $","inlinerMath-JHiC7V8Qr",{"__TSPROSE_proseElement":215,"schema":2526,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2528,"data":2529,"storageKey":1564,"id":2530},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1563},"inlinerMath-AM9HQ4pvg-2",{"__TSPROSE_proseElement":215,"schema":2532,"data":2533},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Both are nice squares, so we can see right away that ",{"__TSPROSE_proseElement":215,"schema":2535,"data":2536,"storageKey":2538,"id":2539},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2537},"a = 3x","$ a = 3x $","inlinerMath-gOFjcUN5E",{"__TSPROSE_proseElement":215,"schema":2541,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2543,"data":2544,"storageKey":1766,"id":2545},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1765},"inlinerMath-jT8WRtcC5-1",{"__TSPROSE_proseElement":215,"schema":2547,"data":1411},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2549,"data":2550,"storageKey":2552,"id":2553},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2551},"-2ab = -2 \\cdot 3x \\cdot 1 = -6x","$ -2ab = -2 \\cdot 3x \\cdot 1 = -6x $","inlinerMath-bd0EntIKb",{"__TSPROSE_proseElement":215,"schema":2555,"data":2556},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," and it matches. So we write:","paragraph-4CV7UyT1a",{"__TSPROSE_proseElement":215,"schema":2559,"data":2560,"storageKey":2562,"id":2563},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2561,"freeze":219},"9x^2 - 6x + 1 = \\underset{a^2}{(3x)^2} - 2 \\cdot \\underset{a}{3x} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = (3x - 1)^2","$$ 9x^2 - 6x + 1 = \\underset{a^2}{(3x)^2} - 2 \\cdot \\underset{a}{3x} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = (3x - 1)^2 $$","blockMath-y8LnumBRS",{"__TSPROSE_proseElement":215,"schema":2565,"data":2566,"children":2567},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},"Using the Middle Term",[2568,2656],{"__TSPROSE_proseElement":215,"schema":2569,"children":2570,"id":2655},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2571,2573,2579,2581,2585,2587,2593,2596,2602,2604,2608,2611,2615,2618,2624,2626,2630,2632,2636,2638,2644,2646,2652],{"__TSPROSE_proseElement":215,"schema":2572,"data":1437},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2574,"data":2575,"storageKey":2577,"id":2578},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2576},"-6x","$ -6x $","inlinerMath-MG3Leuh7I",{"__TSPROSE_proseElement":215,"schema":2580,"data":1446},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2582,"data":2583,"storageKey":2308,"id":2584},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2307},"inlinerMath-02bjIi1Ry-1",{"__TSPROSE_proseElement":215,"schema":2586,"data":1453},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2588,"data":2589,"storageKey":2591,"id":2592},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2590},"3x","$ 3x $","inlinerMath-WYRLesvjE",{"__TSPROSE_proseElement":215,"schema":2594,"data":2595},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". This product must be split into two factors whose squares match the outer terms ",{"__TSPROSE_proseElement":215,"schema":2597,"data":2598,"storageKey":2600,"id":2601},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2599},"9x^2","$ 9x^2 $","inlinerMath-oeT1YYBtQ",{"__TSPROSE_proseElement":215,"schema":2603,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2605,"data":2606,"storageKey":1020,"id":2607},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":385},"inlinerMath-GWJ3eIxvK-3",{"__TSPROSE_proseElement":215,"schema":2609,"data":2610},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Since ",{"__TSPROSE_proseElement":215,"schema":2612,"data":2613,"storageKey":1538,"id":2614},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1537},"inlinerMath-ooHpGun5c-4",{"__TSPROSE_proseElement":215,"schema":2616,"data":2617},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", we artificially include a factor of one: ",{"__TSPROSE_proseElement":215,"schema":2619,"data":2620,"storageKey":2622,"id":2623},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2621},"3x = 3x \\cdot 1","$ 3x = 3x \\cdot 1 $","inlinerMath-nbioH3US9",{"__TSPROSE_proseElement":215,"schema":2625,"data":1488},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2627,"data":2628,"storageKey":2538,"id":2629},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2537},"inlinerMath-gOFjcUN5E-1",{"__TSPROSE_proseElement":215,"schema":2631,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2633,"data":2634,"storageKey":1766,"id":2635},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":1765},"inlinerMath-jT8WRtcC5-2",{"__TSPROSE_proseElement":215,"schema":2637,"data":1668},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2639,"data":2640,"storageKey":2642,"id":2643},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2641},"9x^2 = (3x)^2 = a^2","$ 9x^2 = (3x)^2 = a^2 $","inlinerMath-BPY4xTos9",{"__TSPROSE_proseElement":215,"schema":2645,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2647,"data":2648,"storageKey":2650,"id":2651},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2649},"1 = 1^2 = b^2","$ 1 = 1^2 = b^2 $","inlinerMath-BRasBzsqn",{"__TSPROSE_proseElement":215,"schema":2653,"data":2654},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Everything matches, so we write the square of a difference:","paragraph-2M1PIaNFI",{"__TSPROSE_proseElement":215,"schema":2657,"data":2658,"storageKey":2562,"id":2659},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2561,"freeze":219},"blockMath-y8LnumBRS-1",{"__TSPROSE_proseElement":215,"schema":2661,"data":2662,"children":2663},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[2664,2679,2685,2695,2703],{"__TSPROSE_proseElement":215,"schema":2665,"children":2666},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[2667,2673],{"__TSPROSE_proseElement":215,"schema":2668,"children":2669,"id":2672},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2670],{"__TSPROSE_proseElement":215,"schema":2671,"data":2332},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-qYJVaoVtK-1",{"__TSPROSE_proseElement":215,"schema":2674,"data":2675,"storageKey":2677,"id":2678},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2676,"freeze":219},"(-7 + 2a)^2","$$ (-7 + 2a)^2 $$","blockMath-UAZ3MmPrY",{"__TSPROSE_proseElement":215,"schema":2680,"data":2681},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":2682},{"__ERUDIT_CHECK":215,"name":835,"data":2683},{"expr":2684},"4a^2 - 28a + 49",{"__TSPROSE_proseElement":215,"schema":2686,"children":2687},{"name":1009,"type":218,"linkable":219,"__TSPROSE_schema":215},[2688],{"__TSPROSE_proseElement":215,"schema":2689,"children":2690,"id":2694},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2691],{"__TSPROSE_proseElement":215,"schema":2692,"data":2693},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Use the fact that changing the order of addends does not change a sum.","paragraph-eYPBu4RpC",{"__TSPROSE_proseElement":215,"schema":2696,"children":2697},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[2698],{"__TSPROSE_proseElement":215,"schema":2699,"data":2700,"storageKey":2701,"id":2702},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2684,"freeze":219},"$$ 4a^2 - 28a + 49 $$","blockMath-UKdlYItNi",{"__TSPROSE_proseElement":215,"schema":2704,"children":2705},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[2706,2713,2719,2756,2762,2768],{"__TSPROSE_proseElement":215,"schema":2707,"children":2708,"id":2712},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2709],{"__TSPROSE_proseElement":215,"schema":2710,"data":2711},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"This sum has an unusual form. The minus sign is not in the middle but at the beginning. Swap the first and second terms (the sum does not change) so it is easier to compare with the square-of-a-difference formula:","paragraph-yU7hAKPve",{"__TSPROSE_proseElement":215,"schema":2714,"data":2715,"storageKey":2717,"id":2718},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2716,"freeze":219},"(-7 + 2a)^2 = (2a - 7)^2","$$ (-7 + 2a)^2 = (2a - 7)^2 $$","blockMath-vPY8ftQWb",{"__TSPROSE_proseElement":215,"schema":2720,"children":2721,"id":2755},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2722,2725,2729,2732,2738,2740,2744,2746,2752],{"__TSPROSE_proseElement":215,"schema":2723,"data":2724},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Now ",{"__TSPROSE_proseElement":215,"schema":2726,"data":2727,"storageKey":719,"id":2728},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-10",{"__TSPROSE_proseElement":215,"schema":2730,"data":2731},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," in the formula is played by ",{"__TSPROSE_proseElement":215,"schema":2733,"data":2734,"storageKey":2736,"id":2737},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2735},"2a","$ 2a $","inlinerMath-uofr24ZDY",{"__TSPROSE_proseElement":215,"schema":2739,"data":2378},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2741,"data":2742,"storageKey":727,"id":2743},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-10",{"__TSPROSE_proseElement":215,"schema":2745,"data":877},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2747,"data":2748,"storageKey":2750,"id":2751},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2749},"7","$ 7 $","inlinerMath-sYh3uEvdQ",{"__TSPROSE_proseElement":215,"schema":2753,"data":2754},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". From left to right, find the square of the first term, the negative of twice the product of the first and second terms, and the square of the second:","paragraph-IwftS8vB0",{"__TSPROSE_proseElement":215,"schema":2757,"data":2758,"storageKey":2760,"id":2761},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2759,"freeze":219},"(2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49","$$ (2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49 $$","blockMath-iIPFwePt6",{"__TSPROSE_proseElement":215,"schema":2763,"children":2764,"id":2767},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2765],{"__TSPROSE_proseElement":215,"schema":2766,"data":2407},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-xv2qdT6ML-1",{"__TSPROSE_proseElement":215,"schema":2769,"data":2770,"storageKey":2772,"id":2773},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2771,"freeze":219},"(-7 + 2a)^2 = 4a^2 - 28a + 49","$$ (-7 + 2a)^2 = 4a^2 - 28a + 49 $$","blockMath-CQRD6g7ty",{"__TSPROSE_proseElement":215,"schema":2775,"data":2776,"children":2777},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[2778,2793,2799,2809,2817],{"__TSPROSE_proseElement":215,"schema":2779,"children":2780},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[2781,2787],{"__TSPROSE_proseElement":215,"schema":2782,"children":2783,"id":2786},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2784],{"__TSPROSE_proseElement":215,"schema":2785,"data":2468},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-MudZgQrHU-1",{"__TSPROSE_proseElement":215,"schema":2788,"data":2789,"storageKey":2791,"id":2792},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2790,"freeze":219},"- 12k + 4k^2 + 9","$$ - 12k + 4k^2 + 9 $$","blockMath-6wHlw0P5m",{"__TSPROSE_proseElement":215,"schema":2794,"data":2795},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":2796},{"__ERUDIT_CHECK":215,"name":835,"data":2797},{"expr":2798},"(2k - 3)^2",{"__TSPROSE_proseElement":215,"schema":2800,"children":2801},{"name":1009,"type":218,"linkable":219,"__TSPROSE_schema":215},[2802],{"__TSPROSE_proseElement":215,"schema":2803,"children":2804,"id":2808},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2805],{"__TSPROSE_proseElement":215,"schema":2806,"data":2807},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Reorder the terms so they match the order in the square-of-a-difference formula.","paragraph-uMbxwUl3s",{"__TSPROSE_proseElement":215,"schema":2810,"children":2811},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[2812],{"__TSPROSE_proseElement":215,"schema":2813,"data":2814,"storageKey":2815,"id":2816},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2798,"freeze":219},"$$ (2k - 3)^2 $$","blockMath-pruShJQyn",{"__TSPROSE_proseElement":215,"schema":2818,"children":2819},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[2820,2827,2833,2888],{"__TSPROSE_proseElement":215,"schema":2821,"children":2822,"id":2826},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2823],{"__TSPROSE_proseElement":215,"schema":2824,"data":2825},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"This sum has an unusual form: the pure squares are not at the ends. Reorder the terms (the sum does not change) so the mixed term ends up in the middle:","paragraph-X7Mgfd2Yz",{"__TSPROSE_proseElement":215,"schema":2828,"data":2829,"storageKey":2831,"id":2832},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2830,"freeze":219},"- 12k + 4k^2 + 9 = 4k^2 - 12k + 9","$$ - 12k + 4k^2 + 9 = 4k^2 - 12k + 9 $$","blockMath-DtE8x3QKN",{"__TSPROSE_proseElement":215,"schema":2834,"data":2512,"children":2835},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},[2836,2882],{"__TSPROSE_proseElement":215,"schema":2837,"children":2838,"id":2881},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2839,2841,2847,2849,2855,2857,2863,2865,2871,2873,2879],{"__TSPROSE_proseElement":215,"schema":2840,"data":1377},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2842,"data":2843,"storageKey":2845,"id":2846},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2844},"4k^2 = (2k)^2","$ 4k^2 = (2k)^2 $","inlinerMath-JJuiBLiUp",{"__TSPROSE_proseElement":215,"schema":2848,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2850,"data":2851,"storageKey":2853,"id":2854},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2852},"9 = 3^2","$ 9 = 3^2 $","inlinerMath-WGLl6WNtY",{"__TSPROSE_proseElement":215,"schema":2856,"data":2533},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2858,"data":2859,"storageKey":2861,"id":2862},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2860},"a = 2k","$ a = 2k $","inlinerMath-uOpB6qvj4",{"__TSPROSE_proseElement":215,"schema":2864,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2866,"data":2867,"storageKey":2869,"id":2870},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2868},"b = 3","$ b = 3 $","inlinerMath-FrcWrXeOp",{"__TSPROSE_proseElement":215,"schema":2872,"data":1411},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2874,"data":2875,"storageKey":2877,"id":2878},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2876},"-2ab = -2 \\cdot 2k \\cdot 3 = -12k","$ -2ab = -2 \\cdot 2k \\cdot 3 = -12k $","inlinerMath-tpKMpxWgT",{"__TSPROSE_proseElement":215,"schema":2880,"data":2556},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-XPymXjAMp",{"__TSPROSE_proseElement":215,"schema":2883,"data":2884,"storageKey":2886,"id":2887},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2885,"freeze":219},"4k^2 - 12k + 9 = \\underset{a^2}{(2k)^2} - 2 \\cdot \\underset{a}{2k} \\cdot \\underset{b}{3} + \\underset{b^2}{3^2} = (2k - 3)^2","$$ 4k^2 - 12k + 9 = \\underset{a^2}{(2k)^2} - 2 \\cdot \\underset{a}{2k} \\cdot \\underset{b}{3} + \\underset{b^2}{3^2} = (2k - 3)^2 $$","blockMath-S0jbGcHVk",{"__TSPROSE_proseElement":215,"schema":2889,"data":2566,"children":2890},{"name":1052,"type":218,"linkable":219,"__TSPROSE_schema":215},[2891,2972],{"__TSPROSE_proseElement":215,"schema":2892,"children":2893,"id":2971},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2894,2896,2902,2904,2908,2910,2916,2918,2924,2926,2932,2935,2941,2943,2947,2949,2953,2955,2961,2963,2969],{"__TSPROSE_proseElement":215,"schema":2895,"data":1437},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2897,"data":2898,"storageKey":2900,"id":2901},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2899},"-12k","$ -12k $","inlinerMath-WgJ1mg3NQ",{"__TSPROSE_proseElement":215,"schema":2903,"data":1446},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2905,"data":2906,"storageKey":2308,"id":2907},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2307},"inlinerMath-02bjIi1Ry-2",{"__TSPROSE_proseElement":215,"schema":2909,"data":1453},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2911,"data":2912,"storageKey":2914,"id":2915},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2913},"6k","$ 6k $","inlinerMath-OSxZQEvnW",{"__TSPROSE_proseElement":215,"schema":2917,"data":2595},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2919,"data":2920,"storageKey":2922,"id":2923},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2921},"4k^2","$ 4k^2 $","inlinerMath-OGjRamVXQ",{"__TSPROSE_proseElement":215,"schema":2925,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2927,"data":2928,"storageKey":2930,"id":2931},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2929},"9","$ 9 $","inlinerMath-Ss7zKstt6",{"__TSPROSE_proseElement":215,"schema":2933,"data":2934},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Here that is immediate: ",{"__TSPROSE_proseElement":215,"schema":2936,"data":2937,"storageKey":2939,"id":2940},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2938},"6k = 2k \\cdot 3","$ 6k = 2k \\cdot 3 $","inlinerMath-pRE4jTIIT",{"__TSPROSE_proseElement":215,"schema":2942,"data":1488},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2944,"data":2945,"storageKey":2861,"id":2946},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2860},"inlinerMath-uOpB6qvj4-1",{"__TSPROSE_proseElement":215,"schema":2948,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2950,"data":2951,"storageKey":2869,"id":2952},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2868},"inlinerMath-FrcWrXeOp-1",{"__TSPROSE_proseElement":215,"schema":2954,"data":1668},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2956,"data":2957,"storageKey":2959,"id":2960},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2958},"(2k)^2 = 4k^2 = a^2","$ (2k)^2 = 4k^2 = a^2 $","inlinerMath-bjOiv16QN",{"__TSPROSE_proseElement":215,"schema":2962,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":2964,"data":2965,"storageKey":2967,"id":2968},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":2966},"3^2 = 9 = b^2","$ 3^2 = 9 = b^2 $","inlinerMath-xD4o3bMWT",{"__TSPROSE_proseElement":215,"schema":2970,"data":2654},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-Iaxz8ETTK",{"__TSPROSE_proseElement":215,"schema":2973,"data":2974,"storageKey":2886,"id":2975},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2885,"freeze":219},"blockMath-S0jbGcHVk-1",{"__TSPROSE_proseElement":215,"schema":2977,"data":2978,"children":2979},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[2980,2995,3001,3010],{"__TSPROSE_proseElement":215,"schema":2981,"children":2982},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[2983,2989],{"__TSPROSE_proseElement":215,"schema":2984,"children":2985,"id":2988},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[2986],{"__TSPROSE_proseElement":215,"schema":2987,"data":2332},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-qYJVaoVtK-2",{"__TSPROSE_proseElement":215,"schema":2990,"data":2991,"storageKey":2993,"id":2994},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":2992,"freeze":219},"\\left( 5y - 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From left to right, find the square of the first term, the negative of twice the product of the first and second terms, and the square of the second term. The numbers are more complicated here, so it is reasonable to compute all three separately on paper. Do not forget to simplify.","paragraph-UU93cM1FU",{"__TSPROSE_proseElement":215,"schema":3050,"data":3051,"storageKey":3053,"id":3054},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3052,"freeze":219},"(5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2","$$ (5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2 $$","blockMath-ZNKwF9cMD",{"__TSPROSE_proseElement":215,"schema":3056,"children":3057,"id":3060},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3058],{"__TSPROSE_proseElement":215,"schema":3059,"data":2407},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-xv2qdT6ML-2",{"__TSPROSE_proseElement":215,"schema":3062,"data":3063,"storageKey":3065,"id":3066},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3064,"freeze":219},"\\left( 5y - 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That gives ",{"__TSPROSE_proseElement":215,"schema":3183,"data":3184,"storageKey":3186,"id":3187},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3185},"\\frac{5m}{2}","$ \\frac{5m}{2} $","inlinerMath-l3IYVzMkX",{"__TSPROSE_proseElement":215,"schema":3189,"data":3190},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Now write this fraction as a product of two factors whose squares match the outer terms ",{"__TSPROSE_proseElement":215,"schema":3192,"data":3193,"storageKey":3195,"id":3196},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3194},"\\frac{1}{4}m^2","$ \\frac{1}{4}m^2 $","inlinerMath-xhICl1aoZ",{"__TSPROSE_proseElement":215,"schema":3198,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3200,"data":3201,"storageKey":3203,"id":3204},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3202},"25","$ 25 $","inlinerMath-iCjF04534",{"__TSPROSE_proseElement":215,"schema":3206,"data":3207},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". We factor it as ",{"__TSPROSE_proseElement":215,"schema":3209,"data":3210,"storageKey":3212,"id":3213},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3211},"\\frac{5m}{2} = \\frac{m}{2} \\cdot 5","$ \\frac{5m}{2} = \\frac{m}{2} \\cdot 5 $","inlinerMath-qax14IJhI",{"__TSPROSE_proseElement":215,"schema":3215,"data":1488},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3217,"data":3218,"storageKey":3132,"id":3219},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3131},"inlinerMath-eUTZMMfdg-1",{"__TSPROSE_proseElement":215,"schema":3221,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3223,"data":3224,"storageKey":3140,"id":3225},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3139},"inlinerMath-pB2WH7ceV-1",{"__TSPROSE_proseElement":215,"schema":3227,"data":1668},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3229,"data":3230,"storageKey":3232,"id":3233},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3231},"\\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2","$ \\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2 $","inlinerMath-3XGUv7kWj",{"__TSPROSE_proseElement":215,"schema":3235,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3237,"data":3238,"storageKey":3240,"id":3241},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3239},"5^2 = 25 = b^2","$ 5^2 = 25 = b^2 $","inlinerMath-tDEurJ8Oh",{"__TSPROSE_proseElement":215,"schema":3243,"data":2654},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-99UAlHyil",{"__TSPROSE_proseElement":215,"schema":3246,"data":3247,"storageKey":3157,"id":3248},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3156,"freeze":219},"blockMath-xBGmZf40t-1","square-diff-examples",{"__TSPROSE_proseElement":215,"schema":3251,"data":3252,"id":3254},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":173,"title":3253},"Applications of the Square and a Difference","applications-of-the-square-and-a-difference",{"__TSPROSE_proseElement":215,"schema":3256,"children":3257,"id":3261},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3258],{"__TSPROSE_proseElement":215,"schema":3259,"data":3260},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Earlier we already mentioned one benefit of these Special Products formulas: they let us quickly rewrite expressions either as sums or as products. That makes complicated expressions easier to simplify. Now it is time to give a few more very useful and concrete examples of how these formulas are used.","paragraph-RDacLvm4H",{"__TSPROSE_proseElement":215,"schema":3263,"data":3264,"id":3266},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":175,"title":3265},"Squaring Numbers Quickly","squaring-numbers-quickly",{"__TSPROSE_proseElement":215,"schema":3268,"children":3269,"id":3349},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3270,3273,3278,3281,3287,3290,3296,3299,3305,3307,3313,3316,3322,3325,3331,3333,3339,3341,3347],{"__TSPROSE_proseElement":215,"schema":3271,"data":3272},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Almost everyone can square any number from ",{"__TSPROSE_proseElement":215,"schema":3274,"data":3275,"storageKey":3276,"id":3277},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":275},"$ 0 $","inlinerMath-Tp4aVVyKv",{"__TSPROSE_proseElement":215,"schema":3279,"data":3280},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," to ",{"__TSPROSE_proseElement":215,"schema":3282,"data":3283,"storageKey":3285,"id":3286},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3284},"10","$ 10 $","inlinerMath-8Y3Z60Vbg",{"__TSPROSE_proseElement":215,"schema":3288,"data":3289},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". That is just the multiplication table: ",{"__TSPROSE_proseElement":215,"schema":3291,"data":3292,"storageKey":3294,"id":3295},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3293},"4^2 = 16","$ 4^2 = 16 $","inlinerMath-xhyKJB6AD",{"__TSPROSE_proseElement":215,"schema":3297,"data":3298},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", ",{"__TSPROSE_proseElement":215,"schema":3300,"data":3301,"storageKey":3303,"id":3304},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3302},"6^2 = 36","$ 6^2 = 36 $","inlinerMath-XrAuT94Y3",{"__TSPROSE_proseElement":215,"schema":3306,"data":3298},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3308,"data":3309,"storageKey":3311,"id":3312},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3310},"9^2 = 81","$ 9^2 = 81 $","inlinerMath-qgJlqOjlm",{"__TSPROSE_proseElement":215,"schema":3314,"data":3315},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". You do not have trouble with the multiplication table, do you? 👀 Students who are strong in math and olympiad contestants often know all squares up to ",{"__TSPROSE_proseElement":215,"schema":3317,"data":3318,"storageKey":3320,"id":3321},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3319},"20","$ 20 $","inlinerMath-pnamxzA0J",{"__TSPROSE_proseElement":215,"schema":3323,"data":3324},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," by heart. For example, ",{"__TSPROSE_proseElement":215,"schema":3326,"data":3327,"storageKey":3329,"id":3330},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3328},"11^2 = 121","$ 11^2 = 121 $","inlinerMath-4kF3KIfxY",{"__TSPROSE_proseElement":215,"schema":3332,"data":3298},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3334,"data":3335,"storageKey":3337,"id":3338},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3336},"15^2 = 225","$ 15^2 = 225 $","inlinerMath-pNgw1XMK1",{"__TSPROSE_proseElement":215,"schema":3340,"data":3298},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3342,"data":3343,"storageKey":3345,"id":3346},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3344},"19^2 = 361","$ 19^2 = 361 $","inlinerMath-2YJjOCZ92",{"__TSPROSE_proseElement":215,"schema":3348,"data":758},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-6A7hawvPq",{"__TSPROSE_proseElement":215,"schema":3351,"children":3352,"id":3387},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3353,3356,3362,3365,3369,3371,3375,3378,3384],{"__TSPROSE_proseElement":215,"schema":3354,"data":3355},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"So it turns out that ",{"__TSPROSE_proseElement":215,"schema":3357,"data":3358,"storageKey":3360,"id":3361},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3359},"99%","$ 99% $","inlinerMath-WuB1oQhHw",{"__TSPROSE_proseElement":215,"schema":3363,"data":3364},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," of people only know square values by heart in the range from ",{"__TSPROSE_proseElement":215,"schema":3366,"data":3367,"storageKey":3276,"id":3368},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":275},"inlinerMath-Tp4aVVyKv-1",{"__TSPROSE_proseElement":215,"schema":3370,"data":3280},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3372,"data":3373,"storageKey":3320,"id":3374},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3319},"inlinerMath-pnamxzA0J-1",{"__TSPROSE_proseElement":215,"schema":3376,"data":3377},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". But what if you need to square a somewhat larger number, say something ",{"__TSPROSE_proseElement":215,"schema":3379,"data":3380,"storageKey":3382,"id":3383},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3381},"> 15","$ > 15 $","inlinerMath-LF0AT1QkX",{"__TSPROSE_proseElement":215,"schema":3385,"data":3386},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", quickly and without doing long multiplication?","paragraph-4wRvWwlj4",{"__TSPROSE_proseElement":215,"schema":3389,"children":3390,"id":3394},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3391],{"__TSPROSE_proseElement":215,"schema":3392,"data":3393},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"In today's digital world that is, of course, not much of a problem: you take out your phone, open a calculator, and compute it. But what if you do not have your phone, for example during an exam, or you would have to go get it? That is where the square-of-a-sum and square-of-a-difference formulas come to the rescue.","paragraph-eIVniodra",{"__TSPROSE_proseElement":215,"schema":3396,"data":3397,"children":3402,"id":3848},{"name":239,"type":218,"linkable":215,"__TSPROSE_schema":215},{"title":3398,"level":3399,"attributes":3400},"Quick Squares","easy",[3401],"method",[3403,3614,3790],{"__TSPROSE_proseElement":215,"schema":3404,"data":3405,"children":3407},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{"label":3406},"Sum Example",[3408,3432,3438,3455,3478,3490,3591],{"__TSPROSE_proseElement":215,"schema":3409,"children":3410},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[3411],{"__TSPROSE_proseElement":215,"schema":3412,"children":3413,"id":3431},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3414,3417,3423,3425,3428],{"__TSPROSE_proseElement":215,"schema":3415,"data":3416},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Without multiplying directly, square the number ",{"__TSPROSE_proseElement":215,"schema":3418,"data":3419,"storageKey":3421,"id":3422},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3420},"62","$ 62 $","inlinerMath-350cbZacR",{"__TSPROSE_proseElement":215,"schema":3424,"data":758},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3426},{"name":3427,"type":234,"linkable":219,"__TSPROSE_schema":215},"br",{"__TSPROSE_proseElement":215,"schema":3429,"data":3430},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Try to do it 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numbers?","paragraph-v1WyFV4iX",{"__TSPROSE_proseElement":215,"schema":3456,"children":3457},{"name":1009,"type":218,"linkable":219,"__TSPROSE_schema":215},[3458],{"__TSPROSE_proseElement":215,"schema":3459,"children":3460,"id":3477},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3461,3465,3468,3474],{"__TSPROSE_proseElement":215,"schema":3462,"data":3463,"storageKey":3421,"id":3464},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3420},"inlinerMath-350cbZacR-2",{"__TSPROSE_proseElement":215,"schema":3466,"data":3467},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," is very close to the round number ",{"__TSPROSE_proseElement":215,"schema":3469,"data":3470,"storageKey":3472,"id":3473},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3471},"60","$ 60 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",{"__TSPROSE_proseElement":215,"schema":3507,"data":3508,"storageKey":3472,"id":3509},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3471},"inlinerMath-XhWcFzcMl-1",{"__TSPROSE_proseElement":215,"schema":3511,"data":3512},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". More exactly, ",{"__TSPROSE_proseElement":215,"schema":3514,"data":3515,"storageKey":3517,"id":3518},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3516},"62 = 60 + 2","$ 62 = 60 + 2 $","inlinerMath-BJyKLHwE3",{"__TSPROSE_proseElement":215,"schema":3520,"data":3521},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Instead of computing ",{"__TSPROSE_proseElement":215,"schema":3523,"data":3524,"storageKey":3526,"id":3527},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3525},"62^2","$ 62^2 $","inlinerMath-RfYoVLg4B",{"__TSPROSE_proseElement":215,"schema":3529,"data":3530},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," by hand, it is easier to use the square-of-a-sum formula:","paragraph-PRe0BleOn",{"__TSPROSE_proseElement":215,"schema":3533,"data":3534,"storageKey":3536,"id":3537},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3535,"freeze":219},"62^2 = (60 + 2)^2 = 60^2 + 2 \\cdot 60 \\cdot 2 + 2^2 = 3600 + 240 + 4 = 3844","$$ 62^2 = (60 + 2)^2 = 60^2 + 2 \\cdot 60 \\cdot 2 + 2^2 = 3600 + 240 + 4 = 3844 $$","blockMath-L7kqtB2GR",{"__TSPROSE_proseElement":215,"schema":3539,"children":3540,"id":3566},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3541,3544,3548,3550,3554,3557,3563],{"__TSPROSE_proseElement":215,"schema":3542,"data":3543},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"We broke one difficult process into several simple steps, each of which is easy to do mentally: squaring the two easy numbers ",{"__TSPROSE_proseElement":215,"schema":3545,"data":3546,"storageKey":3472,"id":3547},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3471},"inlinerMath-XhWcFzcMl-2",{"__TSPROSE_proseElement":215,"schema":3549,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3551,"data":3552,"storageKey":525,"id":3553},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"inlinerMath-z9Yar9waf-8",{"__TSPROSE_proseElement":215,"schema":3555,"data":3556},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and doing the very simple multiplication ",{"__TSPROSE_proseElement":215,"schema":3558,"data":3559,"storageKey":3561,"id":3562},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3560},"2 \\cdot 60 \\cdot 2","$ 2 \\cdot 60 \\cdot 2 $","inlinerMath-ddaw3xU0P",{"__TSPROSE_proseElement":215,"schema":3564,"data":3565},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Then we just add.","paragraph-5DsA5FoWd",{"__TSPROSE_proseElement":215,"schema":3568,"children":3569,"id":3590},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3570,3573,3579,3581,3587],{"__TSPROSE_proseElement":215,"schema":3571,"data":3572},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"There is another convenience here: the first two terms in the expansion always end in zero, as in the example above where we got ",{"__TSPROSE_proseElement":215,"schema":3574,"data":3575,"storageKey":3577,"id":3578},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3576},"3600","$ 3600 $","inlinerMath-MEvdagxpW",{"__TSPROSE_proseElement":215,"schema":3580,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3582,"data":3583,"storageKey":3585,"id":3586},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3584},"240","$ 240 $","inlinerMath-7jme5924B",{"__TSPROSE_proseElement":215,"schema":3588,"data":3589},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Numbers like that are easier to add and subtract.","paragraph-ZrsYpvC0X",{"__TSPROSE_proseElement":215,"schema":3592,"children":3593},{"name":2417,"type":218,"linkable":219,"__TSPROSE_schema":215},[3594,3608],{"__TSPROSE_proseElement":215,"schema":3595,"children":3596,"id":3607},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3597,3600,3604],{"__TSPROSE_proseElement":215,"schema":3598,"data":3599},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The idea is this: if the number you want to square is a little larger, within about ",{"__TSPROSE_proseElement":215,"schema":3601,"data":3602,"storageKey":882,"id":3603},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":881},"inlinerMath-MAD3UaxRg-1",{"__TSPROSE_proseElement":215,"schema":3605,"data":3606},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", than the nearest round number ending in zero, then you can write it as a sum and use the square-of-a-sum formula. 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",{"__TSPROSE_proseElement":215,"schema":3746,"data":3747,"storageKey":3678,"id":3748},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3677},"inlinerMath-qW3PO0118-2",{"__TSPROSE_proseElement":215,"schema":3750,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3752,"data":3753,"storageKey":525,"id":3754},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"inlinerMath-z9Yar9waf-9",{"__TSPROSE_proseElement":215,"schema":3756,"data":3556},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":3758,"data":3759,"storageKey":3761,"id":3762},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3760},"2 \\cdot 50 \\cdot 2","$ 2 \\cdot 50 \\cdot 2 $","inlinerMath-as73CCTZi",{"__TSPROSE_proseElement":215,"schema":3764,"data":3765},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". 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If it is a little ",{"__TSPROSE_proseElement":215,"schema":3856,"data":3857,"children":3858,"id":3862},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[3859],{"__TSPROSE_proseElement":215,"schema":3860,"data":3861},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"larger","emphasis-sof9Eabrc",{"__TSPROSE_proseElement":215,"schema":3864,"data":3865},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," than that round number, write it as a sum and use the square-of-a-sum formula. If it is a little ",{"__TSPROSE_proseElement":215,"schema":3867,"data":3868,"children":3869,"id":3873},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[3870],{"__TSPROSE_proseElement":215,"schema":3871,"data":3872},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"smaller","emphasis-Xd2mNJS0X",{"__TSPROSE_proseElement":215,"schema":3875,"data":3876},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", write it as a difference and use the square-of-a-difference formula. In both cases the expansion gives three terms, and ",{"__TSPROSE_proseElement":215,"schema":3878,"data":3879,"children":3880,"id":3884},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[3881],{"__TSPROSE_proseElement":215,"schema":3882,"data":3883},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"the first two always end in zero","emphasis-yuRzYuNpY",{"__TSPROSE_proseElement":215,"schema":3886,"data":3887},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", which makes the final addition easier.","paragraph-hfCItNMOJ",{"__TSPROSE_proseElement":215,"schema":3890,"data":3891,"id":3893},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":175,"title":3892},"Solving Quadratic Equations","solving-quadratic-equations",{"__TSPROSE_proseElement":215,"schema":3895,"children":3896,"id":3900},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3897],{"__TSPROSE_proseElement":215,"schema":3898,"data":3899},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Many different processes in everyday life, physics, mathematics, and other sciences reduce to equations that look a lot like these:","paragraph-g8XX2Yff6",{"__TSPROSE_proseElement":215,"schema":3902,"data":3903,"storageKey":3905,"id":3906},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3904,"freeze":219},"x^2 + 2x + 1 = 0 >>{big} 36 - 18t + 9t^2 = 0 >>{big} 4z^2 + 48z + 144 = 0","$$ x^2 + 2x + 1 = 0 >>{big} 36 - 18t + 9t^2 = 0 >>{big} 4z^2 + 48z + 144 = 0 $$","blockMath-ZdCgLVm87",{"__TSPROSE_proseElement":215,"schema":3908,"children":3909,"id":3922},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3910,3913,3919],{"__TSPROSE_proseElement":215,"schema":3911,"data":3912},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"These are all called quadratic equations, and the task is to determine which number is hidden behind the variable. It has to be a number that makes the left-hand side equal zero when substituted in place of the variable (",{"__TSPROSE_proseElement":215,"schema":3914,"data":3915,"storageKey":3917,"id":3918},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3916},"0 = 0","$ 0 = 0 $","inlinerMath-auVBecTzm",{"__TSPROSE_proseElement":215,"schema":3920,"data":3921},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"), otherwise the equation will not be true.","paragraph-F1Eq7bWAr",{"__TSPROSE_proseElement":215,"schema":3924,"children":3925,"id":3929},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3926],{"__TSPROSE_proseElement":215,"schema":3927,"data":3928},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"So what numbers are they? You could only guess and plug in different values at random. But everything becomes much simpler if you notice that the left-hand sides are already expanded forms of the square of a sum or the square of a difference. Factor them back into parentheses and we get:","paragraph-uejfYonxU",{"__TSPROSE_proseElement":215,"schema":3931,"data":3932,"storageKey":3934,"id":3935},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":3933,"freeze":219},"(x+1)^2 = 0 >>{big} (6 - 3t)^2 = 0 >>{big} (2z + 12)^2 = 0","$$ (x+1)^2 = 0 >>{big} (6 - 3t)^2 = 0 >>{big} (2z + 12)^2 = 0 $$","blockMath-PjWSJDY4N",{"__TSPROSE_proseElement":215,"schema":3937,"children":3938,"id":4028},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[3939,3942,3948,3951,3955,3958,3964,3967,3971,3974,3980,3983,3987,3989,3993,3995,3999,4002,4008,4011,4017,4020,4026],{"__TSPROSE_proseElement":215,"schema":3940,"data":3941},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Now it is easy. 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Right: we should substitute ",{"__TSPROSE_proseElement":215,"schema":3959,"data":3960,"storageKey":3962,"id":3963},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3961},"-1","$ -1 $","inlinerMath-DLMG5r2qd",{"__TSPROSE_proseElement":215,"schema":3965,"data":3966},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," for ",{"__TSPROSE_proseElement":215,"schema":3968,"data":3969,"storageKey":3946,"id":3970},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":3945},"inlinerMath-TLSwLL8iH-1",{"__TSPROSE_proseElement":215,"schema":3972,"data":3973},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". 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We should substitute ",{"__TSPROSE_proseElement":215,"schema":4021,"data":4022,"storageKey":4024,"id":4025},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4023},"-6","$ -6 $","inlinerMath-6kVj401tO",{"__TSPROSE_proseElement":215,"schema":4027,"data":758},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-KovTCUJXA",{"__TSPROSE_proseElement":215,"schema":4030,"data":4031,"storageKey":4033,"id":4034},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":4032,"freeze":219},"(\\underset{x}{-1} + 1)^2 = 0 >>{big} (6 - 3 \\cdot \\underset{t}{2})^2 = 0 >>{big} (2 \\cdot \\underset{z}{-6} + 12)^2 = 0","$$ (\\underset{x}{-1} + 1)^2 = 0 >>{big} (6 - 3 \\cdot \\underset{t}{2})^2 = 0 >>{big} (2 \\cdot \\underset{z}{-6} + 12)^2 = 0 $$","blockMath-08Z3IgRTI",{"__TSPROSE_proseElement":215,"schema":4036,"children":4037,"id":4041},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4038],{"__TSPROSE_proseElement":215,"schema":4039,"data":4040},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"It may seem as if we have solved just a few mathematical puzzles, but in fact these variables can stand for the number of packs of cookies bought, the running time of a car engine, and other real quantities.","paragraph-gwjPmYtbi",{"__TSPROSE_proseElement":215,"schema":4043,"data":4045,"storageKey":4047,"id":4048},{"name":4044,"type":218,"linkable":215,"__TSPROSE_schema":215},"referenceBlock",{"label":4046},"The process of factoring an expanded expression into a square by using the square-of-a-sum or square-of-a-difference formula is called completing the square and is a universal method for solving any quadratic equation.","\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square","referenceBlock-6Srd8sb5I",{"__TSPROSE_proseElement":215,"schema":4050,"data":4051,"id":4053},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":173,"title":4052},"Difference of Squares","difference-of-squares",{"__TSPROSE_proseElement":215,"schema":4055,"children":4056,"id":4080},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4057,4060,4066,4069,4077],{"__TSPROSE_proseElement":215,"schema":4058,"data":4059},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"There is one more very useful and simple formula you need to know by heart. It is called the difference of squares and is written exactly the way it is read: ",{"__TSPROSE_proseElement":215,"schema":4061,"data":4062,"storageKey":4064,"id":4065},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4063},"a^2 - b^2","$ a^2 - b^2 $","inlinerMath-T1YOXnC6X",{"__TSPROSE_proseElement":215,"schema":4067,"data":4068},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," (the difference of ",{"__TSPROSE_proseElement":215,"schema":4070,"data":4071,"children":4072,"id":4076},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[4073],{"__TSPROSE_proseElement":215,"schema":4074,"data":4075},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"two","emphasis-LrgCFxe45",{"__TSPROSE_proseElement":215,"schema":4078,"data":4079},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," squares, the difference of two numbers squared).","paragraph-BZMuBb8Wt",{"__TSPROSE_proseElement":215,"schema":4082,"children":4083,"id":4131},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4084,4087,4091,4094,4098,4101,4105,4108,4112,4115,4119,4122,4128],{"__TSPROSE_proseElement":215,"schema":4085,"data":4086},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"It is not obvious at first glance what the difference of squares ",{"__TSPROSE_proseElement":215,"schema":4088,"data":4089,"storageKey":4064,"id":4090},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4063},"inlinerMath-T1YOXnC6X-1",{"__TSPROSE_proseElement":215,"schema":4092,"data":4093},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," equals, so it is more convenient to start with a geometric derivation. Take a larger square with area ",{"__TSPROSE_proseElement":215,"schema":4095,"data":4096,"storageKey":736,"id":4097},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":735},"inlinerMath-eiXlIGTfO-1",{"__TSPROSE_proseElement":215,"schema":4099,"data":4100},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," and cut out a smaller square with area ",{"__TSPROSE_proseElement":215,"schema":4102,"data":4103,"storageKey":754,"id":4104},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":753},"inlinerMath-tBkcHvoEV-1",{"__TSPROSE_proseElement":215,"schema":4106,"data":4107},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Horizontally the length stays ",{"__TSPROSE_proseElement":215,"schema":4109,"data":4110,"storageKey":719,"id":4111},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-12",{"__TSPROSE_proseElement":215,"schema":4113,"data":4114},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", but the height decreases by ",{"__TSPROSE_proseElement":215,"schema":4116,"data":4117,"storageKey":727,"id":4118},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-12",{"__TSPROSE_proseElement":215,"schema":4120,"data":4121},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", so it becomes ",{"__TSPROSE_proseElement":215,"schema":4123,"data":4124,"storageKey":4126,"id":4127},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4125},"a - b","$ a - b $","inlinerMath-zpwuNTMuV",{"__TSPROSE_proseElement":215,"schema":4129,"data":4130},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". 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To do that, just expand the brackets in any way you like, for example with FOIL:","paragraph-iMpteIpNW",{"__TSPROSE_proseElement":215,"schema":4173,"data":4174,"storageKey":4176,"id":4177},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":4175,"freeze":219},"(a+b)(a-b) = a \\cdot a - a \\cdot b + b \\cdot a - b \\cdot b = \\boxed{a^2 - b^2}","$$ (a+b)(a-b) = a \\cdot a - a \\cdot b + b \\cdot a - b \\cdot b = \\boxed{a^2 - b^2} $$","blockMath-wMfvEsRpC",{"__TSPROSE_proseElement":215,"schema":4179},{"name":4180,"type":218,"linkable":219,"__TSPROSE_schema":215},"hr",{"__TSPROSE_proseElement":215,"schema":4182,"children":4183,"id":4187},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4184],{"__TSPROSE_proseElement":215,"schema":4185,"data":4186},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Going the other way is a bit trickier. We need to artificially add and immediately subtract a term, then factor out the common factors:","paragraph-pfAYhN4oz",{"__TSPROSE_proseElement":215,"schema":4189,"data":4190,"storageKey":4192,"id":4193},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":4191,"freeze":219},"a^2 - b^2 = a^2 + \\underbrace{ab - ab}_{\\text{Added and subtracted}} - b^2 = a(a+b) - b(a+b) = \\boxed{(a+b)(a-b)}","$$ a^2 - b^2 = a^2 + \\underbrace{ab - ab}_{\\text{Added and subtracted}} - b^2 = a(a+b) - b(a+b) = \\boxed{(a+b)(a-b)} $$","blockMath-5h8GbPyEJ","diff-of-squares",{"__TSPROSE_proseElement":215,"schema":4196,"data":4197,"children":4199,"id":4268},{"name":2222,"type":218,"linkable":215,"__TSPROSE_schema":215},{"title":4198,"layout":535},"Difference of Squares ≠ Square of a Difference!",[4200],{"__TSPROSE_proseElement":215,"schema":4201,"children":4202},{"name":2228,"type":218,"linkable":219,"__TSPROSE_schema":215},[4203],{"__TSPROSE_proseElement":215,"schema":4204,"children":4205,"id":4267},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4206,4209,4217,4220,4228,4231,4235,4238,4246,4249,4257,4259,4265],{"__TSPROSE_proseElement":215,"schema":4207,"data":4208},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Beginners studying these formulas very often confuse the difference of squares with the square of a difference. Avoiding this mistake is easy -- just think about the name for one extra second. Difference of square ",{"__TSPROSE_proseElement":215,"schema":4210,"data":4211,"children":4212,"id":4216},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628,"accent":215},[4213],{"__TSPROSE_proseElement":215,"schema":4214,"data":4215},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"s","emphasis-hRGmH16Kt",{"__TSPROSE_proseElement":215,"schema":4218,"data":4219},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," means there are ",{"__TSPROSE_proseElement":215,"schema":4221,"data":4222,"children":4223,"id":4227},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628,"accent":215},[4224],{"__TSPROSE_proseElement":215,"schema":4225,"data":4226},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"several squares","emphasis-4GfbQ6hvm",{"__TSPROSE_proseElement":215,"schema":4229,"data":4230},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", so it is ",{"__TSPROSE_proseElement":215,"schema":4232,"data":4233,"storageKey":4064,"id":4234},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4063},"inlinerMath-T1YOXnC6X-2",{"__TSPROSE_proseElement":215,"schema":4236,"data":4237},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". 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0.09y^2 = \\underset{a}{(3x^2z)^2} - \\underset{b}{(0.3y)^2} = (3x^2z + 0.3y)(3x^2z - 0.3y)","$$ 9x^4z^2 - 0.09y^2 = \\underset{a}{(3x^2z)^2} - \\underset{b}{(0.3y)^2} = (3x^2z + 0.3y)(3x^2z - 0.3y) $$","blockMath-SQrZiyBBQ","diff-of-squares-examples",{"__TSPROSE_proseElement":215,"schema":4680,"data":4681,"id":4683},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":173,"title":4682},"Cube of a Sum and a Difference","cube-of-a-sum-and-a-difference",{"__TSPROSE_proseElement":215,"schema":4685,"children":4686,"id":4690},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4687],{"__TSPROSE_proseElement":215,"schema":4688,"data":4689},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Up to this point we have only been playing with squares, meaning the second degree. And we already got three interesting formulas: the square of a sum, the square of a difference, and the difference of squares. Can we go further? Are there convenient formulas for working quickly with cubes, that is, the third degree? Of course there are. You do not have to memorize them, but it is useful at least to get familiar with them.","paragraph-p5gubLkId",{"__TSPROSE_proseElement":215,"schema":4692,"children":4693,"id":4697},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4694],{"__TSPROSE_proseElement":215,"schema":4695,"data":4696},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"There are quite a lot of Special Products in general, but we will only touch the most basic ones here -- the cube of a sum and the cube of a difference. Their names say exactly what they mean. A cube of a sum or a difference means you have a sum or a difference of two numbers, and the whole thing is raised to the third power. The formulas are derived the same way as the square of a sum -- by expanding brackets with FOIL:","paragraph-B1YCzRyYv",{"__TSPROSE_proseElement":215,"schema":4699,"data":4700,"storageKey":4702,"id":4703},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":4701,"freeze":219},"(a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = \\boxed{a^3 + 3a^2b + 3ab^2 + b^3} \\\\ (a-b)^3 = (a-b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3}","$$ (a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = \\boxed{a^3 + 3a^2b + 3ab^2 + b^3} \\\\ (a-b)^3 = (a-b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3} $$","blockMath-7jaM7Hka9",{"__TSPROSE_proseElement":215,"schema":4705,"data":4706,"storageKey":4708,"id":4709},{"name":4044,"type":218,"linkable":215,"__TSPROSE_schema":215},{"label":4707},"Deriving them in the opposite direction, from the expanded form back to the compact one, is already pretty tricky. Tricky enough to make a nice separate challenge problem for you to think about 😈 You can solve it now or leave it until you finish the article.","\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fpractice\u002F$cubeSumDiffFactorization","referenceBlock-xTyElds0Z",{"__TSPROSE_proseElement":215,"schema":4711,"children":4712,"id":4716},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4713],{"__TSPROSE_proseElement":215,"schema":4714,"data":4715},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Both formulas can also be derived geometrically. From the names alone, you can already tell that we are now dealing with three-dimensional figures. And instead of assembling a square, we will assemble a cube. It looks like this:","paragraph-AKL9m61lk",{"__TSPROSE_proseElement":215,"schema":4718,"data":4720,"storageKey":4721,"children":4724,"id":4744},{"name":4719,"type":218,"linkable":215,"__TSPROSE_schema":215},"video",{"src":4721,"autoplay":215,"width":4722,"invert":4723},"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fcube-sum.mp4","380px","light",[4725],{"__TSPROSE_proseElement":215,"schema":4726,"children":4727},{"name":675,"type":234,"linkable":219,"__TSPROSE_schema":215},[4728,4731],{"__TSPROSE_proseElement":215,"schema":4729,"data":4730},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Geometric derivation of the cube of a sum",{"__TSPROSE_proseElement":215,"schema":4732,"children":4734},{"name":4733,"type":234,"linkable":219,"__TSPROSE_schema":215},"captionSecondary",[4735,4738],{"__TSPROSE_proseElement":215,"schema":4736,"data":4737},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Taken from the TikTok channel ",{"__TSPROSE_proseElement":215,"schema":4739,"data":4740,"storageKey":4742,"id":4743},{"name":594,"type":234,"linkable":215,"__TSPROSE_schema":215},{"label":4741},"@complex_math","\u003Clink:external>\u002Fhttps:\u002F\u002Fwww.tiktok.com\u002F@complex_math\u002Fvideo\u002F7358570064724970759","referenceInliner-cQNN2wVfA","video-CWzLsKjni",{"__TSPROSE_proseElement":215,"schema":4746,"children":4747,"id":4803},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[4748,4751,4757,4760,4766,4768,4774,4777,4781,4783,4787,4790,4794,4796,4800],{"__TSPROSE_proseElement":215,"schema":4749,"data":4750},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"This visualization makes it immediately clear where the coefficients ",{"__TSPROSE_proseElement":215,"schema":4752,"data":4753,"storageKey":4755,"id":4756},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4754},"3","$ 3 $","inlinerMath-mqn34wvB0",{"__TSPROSE_proseElement":215,"schema":4758,"data":4759},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," and the expressions of the form ",{"__TSPROSE_proseElement":215,"schema":4761,"data":4762,"storageKey":4764,"id":4765},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4763},"a^2b","$ a^2b $","inlinerMath-GU9lQOdXE",{"__TSPROSE_proseElement":215,"schema":4767,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":4769,"data":4770,"storageKey":4772,"id":4773},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4771},"ab^2","$ ab^2 $","inlinerMath-U6dXU6jrQ",{"__TSPROSE_proseElement":215,"schema":4775,"data":4776},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," come from. 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from the last one we get ",{"__TSPROSE_proseElement":215,"schema":5067,"data":5068,"storageKey":5070,"id":5071},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5069},"b^3 = 8n^3 = (2n)^3","$ b^3 = 8n^3 = (2n)^3 $","inlinerMath-bfgsl3Qkn",{"__TSPROSE_proseElement":215,"schema":5073,"data":1488},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":5075,"data":5076,"storageKey":5078,"id":5079},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5077},"b = 2n","$ b = 2n $","inlinerMath-4O5zIOh3y",{"__TSPROSE_proseElement":215,"schema":5081,"data":5082},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Rewrite each term in a form that clearly matches the cube of a sum formula, label them, and pack it up:","paragraph-W09TuaXxo",{"__TSPROSE_proseElement":215,"schema":5085,"data":5086,"storageKey":5088,"id":5089},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5087,"freeze":219},"\\underbrace{m^3}_{a^3} + 3 \\cdot \\underbrace{m^2}_{a^2} \\cdot \\underbrace{2n}_{b} + 3 \\cdot \\underbrace{m}_{a} \\cdot \\underbrace{(2n)^2}_{b^2} + \\underbrace{(2n)^3}_{b^3} = (m + 2n)^3","$$ \\underbrace{m^3}_{a^3} + 3 \\cdot \\underbrace{m^2}_{a^2} \\cdot \\underbrace{2n}_{b} + 3 \\cdot \\underbrace{m}_{a} \\cdot \\underbrace{(2n)^2}_{b^2} + \\underbrace{(2n)^3}_{b^3} = (m + 2n)^3 $$","blockMath-nD4M5tIIP",{"__TSPROSE_proseElement":215,"schema":5091,"data":5092,"children":5093},{"name":254,"type":218,"linkable":219,"__TSPROSE_schema":215},{},[5094,5109,5115,5124],{"__TSPROSE_proseElement":215,"schema":5095,"children":5096},{"name":259,"type":218,"linkable":219,"__TSPROSE_schema":215},[5097,5103],{"__TSPROSE_proseElement":215,"schema":5098,"children":5099,"id":5102},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5100],{"__TSPROSE_proseElement":215,"schema":5101,"data":5005},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-3PGk5rMkH-1",{"__TSPROSE_proseElement":215,"schema":5104,"data":5105,"storageKey":5107,"id":5108},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5106,"freeze":219},"\\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27}","$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} $$","blockMath-AGVrgitHm",{"__TSPROSE_proseElement":215,"schema":5110,"data":5111},{"name":270,"type":218,"linkable":219,"__TSPROSE_schema":215},{"serializedValidator":5112},{"__ERUDIT_CHECK":215,"name":835,"data":5113},{"expr":5114},"(x\u002F2 - y\u002F3)^3",{"__TSPROSE_proseElement":215,"schema":5116,"children":5117},{"name":278,"type":218,"linkable":219,"__TSPROSE_schema":215},[5118],{"__TSPROSE_proseElement":215,"schema":5119,"data":5120,"storageKey":5122,"id":5123},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5121,"freeze":219},"\\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3","$$ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$","blockMath-bBCuNpUCR",{"__TSPROSE_proseElement":215,"schema":5125,"children":5126},{"name":287,"type":218,"linkable":219,"__TSPROSE_schema":215},[5127,5178],{"__TSPROSE_proseElement":215,"schema":5128,"children":5129,"id":5177},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5130,5132,5136,5138,5142,5144,5150,5152,5158,5160,5166,5168,5174],{"__TSPROSE_proseElement":215,"schema":5131,"data":5035},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":5133,"data":5134,"storageKey":719,"id":5135},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":718},"inlinerMath-YqDfaaRXA-20",{"__TSPROSE_proseElement":215,"schema":5137,"data":330},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":5139,"data":5140,"storageKey":727,"id":5141},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":726},"inlinerMath-LTS8nNQCZ-20",{"__TSPROSE_proseElement":215,"schema":5143,"data":5048},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":5145,"data":5146,"storageKey":5148,"id":5149},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5147},"a^3 = x^3\u002F8 = (x\u002F2)^3","$ a^3 = x^3\u002F8 = (x\u002F2)^3 $","inlinerMath-YK5vQm7F8",{"__TSPROSE_proseElement":215,"schema":5151,"data":1488},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":5153,"data":5154,"storageKey":5156,"id":5157},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5155},"a = x\u002F2","$ a = x\u002F2 $","inlinerMath-6CbpESr7U",{"__TSPROSE_proseElement":215,"schema":5159,"data":5065},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":5161,"data":5162,"storageKey":5164,"id":5165},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5163},"b^3 = y^3\u002F27 = (y\u002F3)^3","$ b^3 = y^3\u002F27 = (y\u002F3)^3 $","inlinerMath-D7ku1JQ0R",{"__TSPROSE_proseElement":215,"schema":5167,"data":1488},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},{"__TSPROSE_proseElement":215,"schema":5169,"data":5170,"storageKey":5172,"id":5173},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5171},"b = y\u002F3","$ b = y\u002F3 $","inlinerMath-3xOiTfiAj",{"__TSPROSE_proseElement":215,"schema":5175,"data":5176},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". Rewrite each term in a form that clearly matches the cube of a difference formula, label them, and pack it up:","paragraph-35Iu73agP",{"__TSPROSE_proseElement":215,"schema":5179,"data":5180,"storageKey":5182,"id":5183},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5181,"freeze":219},"\\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} = \\\\ \\underbrace{\\frac{x^3}{8}}_{a^3} - 3 \\cdot \\underbrace{\\frac{x^2}{4}}_{a^2} \\cdot \\underbrace{\\frac{y}{3}}_{b} + 3 \\cdot \\underbrace{\\frac{x}{2}}_{a} \\cdot \\underbrace{\\frac{y^2}{9}}_{b^2} - \\underbrace{\\frac{y^3}{27}}_{b^3} = \\\\ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3","$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} = \\\\ \\underbrace{\\frac{x^3}{8}}_{a^3} - 3 \\cdot \\underbrace{\\frac{x^2}{4}}_{a^2} \\cdot \\underbrace{\\frac{y}{3}}_{b} + 3 \\cdot \\underbrace{\\frac{x}{2}}_{a} \\cdot \\underbrace{\\frac{y^2}{9}}_{b^2} - \\underbrace{\\frac{y^3}{27}}_{b^3} = \\\\ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$","blockMath-4EhtSSJbY","cube-sum-diff-examples",{"__TSPROSE_proseElement":215,"schema":5186,"data":5187,"id":5189},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":173,"title":5188},"How to memorize these formulas?","how-to-memorize-these-formulas",{"__TSPROSE_proseElement":215,"schema":5191,"children":5192,"id":5219},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5193,5196,5202,5205,5209,5212,5216],{"__TSPROSE_proseElement":215,"schema":5194,"data":5195},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"You only need to memorize three formulas by heart: the square of a sum ",{"__TSPROSE_proseElement":215,"schema":5197,"data":5198,"storageKey":5200,"id":5201},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5199},"(a + b)^2","$ (a + b)^2 $","inlinerMath-ECN7SKZxy",{"__TSPROSE_proseElement":215,"schema":5203,"data":5204},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", the square of a difference ",{"__TSPROSE_proseElement":215,"schema":5206,"data":5207,"storageKey":4263,"id":5208},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4262},"inlinerMath-sAdriU9tc-1",{"__TSPROSE_proseElement":215,"schema":5210,"data":5211},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and the difference of squares ",{"__TSPROSE_proseElement":215,"schema":5213,"data":5214,"storageKey":4064,"id":5215},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4063},"inlinerMath-T1YOXnC6X-3",{"__TSPROSE_proseElement":215,"schema":5217,"data":5218},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},". They are everywhere, and you absolutely need to be able to spot them instantly and switch between expanded form and factored form. As for the cube formulas, it is enough to be able to recognize them. Here are a few tips that make all of these formulas easier to remember:","paragraph-q8DfY8jNe",{"__TSPROSE_proseElement":215,"schema":5221,"data":5223,"children":5225,"id":5370},{"name":5222,"type":218,"linkable":215,"__TSPROSE_schema":215},"list",{"type":5224},"unordered",[5226,5262,5322],{"__TSPROSE_proseElement":215,"schema":5227,"children":5229},{"name":5228,"type":218,"linkable":219,"__TSPROSE_schema":215},"listItem",[5230,5242,5249,5255],{"__TSPROSE_proseElement":215,"schema":5231,"children":5232,"id":5241},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5233],{"__TSPROSE_proseElement":215,"schema":5234,"data":5235,"children":5236,"id":5240},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[5237],{"__TSPROSE_proseElement":215,"schema":5238,"data":5239},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Plus-minus sign","emphasis-1NyqRxQyB","paragraph-04bbQkxbo",{"__TSPROSE_proseElement":215,"schema":5243,"children":5244,"id":5248},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5245],{"__TSPROSE_proseElement":215,"schema":5246,"data":5247},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Do not memorize 4 separate formulas: the square of a sum, the square of a difference, the cube of a sum, and the cube of a difference. It is enough to remember 2 formulas if you use the plus-minus sign, because nothing changes except the signs:","paragraph-QOjfwQPVc",{"__TSPROSE_proseElement":215,"schema":5250,"data":5251,"storageKey":5253,"id":5254},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5252,"freeze":219},"(a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3","$$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $$","blockMath-kiNVg4zvS",{"__TSPROSE_proseElement":215,"schema":5256,"children":5257,"id":5261},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5258],{"__TSPROSE_proseElement":215,"schema":5259,"data":5260},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"In sums, all signs are always pluses. In a difference, the minus sign always comes right after the first term in the expansion. In the cube case, it also appears in front of the last term.","paragraph-MY1vgSzhk",{"__TSPROSE_proseElement":215,"schema":5263,"children":5264},{"name":5228,"type":218,"linkable":219,"__TSPROSE_schema":215},[5265,5277,5290],{"__TSPROSE_proseElement":215,"schema":5266,"children":5267,"id":5276},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5268],{"__TSPROSE_proseElement":215,"schema":5269,"data":5270,"children":5271,"id":5275},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[5272],{"__TSPROSE_proseElement":215,"schema":5273,"data":5274},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The difference of squares stands apart","emphasis-kyRJimTRg","paragraph-EKll4URRU",{"__TSPROSE_proseElement":215,"schema":5278,"children":5279,"id":5289},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5280,5283],{"__TSPROSE_proseElement":215,"schema":5281,"data":5282},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Formulas whose names start with the degree, like square and cube, have a similar shape, and they can be derived naturally by expanding brackets ",{"__TSPROSE_proseElement":215,"schema":5284,"data":5285,"storageKey":5287,"id":5288},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5286},"(a+b)^2 = (a+b)(a+b) = \\ldots","$ (a+b)^2 = (a+b)(a+b) = \\ldots $","inlinerMath-E2hB8EXr8","paragraph-xTUIHDIgz",{"__TSPROSE_proseElement":215,"schema":5291,"children":5292,"id":5321},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5293,5296,5302,5305,5309,5312,5318],{"__TSPROSE_proseElement":215,"schema":5294,"data":5295},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"But the difference of squares ",{"__TSPROSE_proseElement":215,"schema":5297,"data":5298,"storageKey":5300,"id":5301},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5299},"a^2-b^2","$ a^2-b^2 $","inlinerMath-Z18OlR0cJ",{"__TSPROSE_proseElement":215,"schema":5303,"data":5304},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," stands apart. First, it factors into brackets with a plus and a minus. Second, from the form ",{"__TSPROSE_proseElement":215,"schema":5306,"data":5307,"storageKey":4064,"id":5308},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4063},"inlinerMath-T1YOXnC6X-4",{"__TSPROSE_proseElement":215,"schema":5310,"data":5311},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," you cannot naturally and explicitly get the product ",{"__TSPROSE_proseElement":215,"schema":5313,"data":5314,"storageKey":5316,"id":5317},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":5315},"(a+b)(a-b)","$ (a+b)(a-b) $","inlinerMath-bZ4wF1qyF",{"__TSPROSE_proseElement":215,"schema":5319,"data":5320},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215}," from the difference itself.","paragraph-bJ4EZXAyz",{"__TSPROSE_proseElement":215,"schema":5323,"children":5324},{"name":5228,"type":218,"linkable":219,"__TSPROSE_schema":215},[5325,5337,5357,5363],{"__TSPROSE_proseElement":215,"schema":5326,"children":5327,"id":5336},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5328],{"__TSPROSE_proseElement":215,"schema":5329,"data":5330,"children":5331,"id":5335},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[5332],{"__TSPROSE_proseElement":215,"schema":5333,"data":5334},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The degree matches the coefficient","emphasis-3bQgNnAYx","paragraph-SOnNSs7tQ",{"__TSPROSE_proseElement":215,"schema":5338,"children":5339,"id":5356},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5340,5343,5347,5350,5354],{"__TSPROSE_proseElement":215,"schema":5341,"data":5342},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"In the square or cube of a sum\u002Fdifference, the degree (second or third) also appears as a coefficient in the expansion. For the square it is ",{"__TSPROSE_proseElement":215,"schema":5344,"data":5345,"storageKey":525,"id":5346},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":524},"inlinerMath-z9Yar9waf-12",{"__TSPROSE_proseElement":215,"schema":5348,"data":5349},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and for the cube it is ",{"__TSPROSE_proseElement":215,"schema":5351,"data":5352,"storageKey":4755,"id":5353},{"name":323,"type":234,"linkable":215,"__TSPROSE_schema":215},{"katex":4754},"inlinerMath-mqn34wvB0-1",{"__TSPROSE_proseElement":215,"schema":5355,"data":1180},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"paragraph-rDkGDBMEF",{"__TSPROSE_proseElement":215,"schema":5358,"data":5359,"storageKey":5361,"id":5362},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5360,"freeze":219},"(a \\pm b)^{\\normalsize\\brand{2}} = a^2 \\pm \\brand{2}ab + b^2 \\\\ (a \\pm b)^{\\normalsize\\brand{3}} = a^3 \\pm \\brand{3}a^2b + \\brand{3}ab^2 \\pm b^3","$$ (a \\pm b)^{\\normalsize\\brand{2}} = a^2 \\pm \\brand{2}ab + b^2 \\\\ (a \\pm b)^{\\normalsize\\brand{3}} = a^3 \\pm \\brand{3}a^2b + \\brand{3}ab^2 \\pm b^3 $$","blockMath-L9UAp9IUH",{"__TSPROSE_proseElement":215,"schema":5364,"children":5365,"id":5369},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5366],{"__TSPROSE_proseElement":215,"schema":5367,"data":5368},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The coefficient is also easy to remember from the geometric derivation. For the square formulas, we build a square, and in the process two rectangles appear. For the cube formulas, we build a cube, and in the process two kinds of three parallelepipeds appear.","paragraph-JZ241jGeh","memorization-tips",{"__TSPROSE_proseElement":215,"schema":5372,"data":5373,"id":5375},{"name":223,"type":218,"linkable":215,"__TSPROSE_schema":215},{"level":173,"title":5374},"Higher powers of Sum and Difference","higher-powers-of-sum-and-difference",{"__TSPROSE_proseElement":215,"schema":5377,"children":5378,"id":5382},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5379],{"__TSPROSE_proseElement":215,"schema":5380,"data":5381},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Have you noticed that the higher the degree, the longer and more complicated these formulas become? Can we keep increasing the degree forever? Can we find the ultimate solution to Freshman's Dream?","paragraph-KyLdy2vNM",{"__TSPROSE_proseElement":215,"schema":5384,"data":5385,"storageKey":5387,"id":5388},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5386,"freeze":219},"(a \\pm b)^1 = a \\pm b \\\\ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 \\\\ \\text{???}","$$ (a \\pm b)^1 = a \\pm b \\\\ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 \\\\ \\text{???} $$","blockMath-nN28RRSij",{"__TSPROSE_proseElement":215,"schema":5390,"children":5391,"id":5406},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5392,5395,5403],{"__TSPROSE_proseElement":215,"schema":5393,"data":5394},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"In fact, we can. There is a very powerful universal formula that automatically produces Special Products for absolutely any degree. It is called the ",{"__TSPROSE_proseElement":215,"schema":5396,"data":5397,"children":5398,"id":5402},{"name":626,"type":234,"linkable":215,"__TSPROSE_schema":215},{"type":628},[5399],{"__TSPROSE_proseElement":215,"schema":5400,"data":5401},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"Binomial Theorem","emphasis-OQnT4q1T4",{"__TSPROSE_proseElement":215,"schema":5404,"data":5405},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},", and it looks like this:","paragraph-Jb8WRNyAb",{"__TSPROSE_proseElement":215,"schema":5408,"data":5409,"storageKey":5411,"id":5412},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5410,"freeze":219},"(a+b)^n = \\sum\\limits_{k=0}^{n} \\binom{n}{k} a^{n-k}b^k, \\quad \\text{where } \\binom{n}{k} = \\frac{n!}{k!(n-k)!}","$$ (a+b)^n = \\sum\\limits_{k=0}^{n} \\binom{n}{k} a^{n-k}b^k, \\quad \\text{where } \\binom{n}{k} = \\frac{n!}{k!(n-k)!} $$","blockMath-2kdJkIhnc",{"__TSPROSE_proseElement":215,"schema":5414,"children":5415,"id":5419},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5416],{"__TSPROSE_proseElement":215,"schema":5417,"data":5418},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"This probably looks shocking. That is normal: the formula really does look intimidating, and deriving it requires some combinatorics, which is the topic where it is usually proved. And yet it does not require any higher mathematics; you can derive it at a regular school level.","paragraph-aG1JsaJNS",{"__TSPROSE_proseElement":215,"schema":5421,"children":5422,"id":5426},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5423],{"__TSPROSE_proseElement":215,"schema":5424,"data":5425},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"So the answer to the question of whether there are infinitely many such formulas is yes. You can keep increasing the degree and get more and more new formulas:","paragraph-YQJ98EnRo",{"__TSPROSE_proseElement":215,"schema":5428,"data":5429,"storageKey":5431,"id":5432},{"name":263,"type":218,"linkable":215,"__TSPROSE_schema":215},{"katex":5430,"freeze":219},"(a \\pm b)^4 = a^4 \\pm 4a^3b + 6a^2b^2 \\pm 4ab^3 + b^4 \\\\ (a \\pm b)^5 = a^5 \\pm 5a^4b + 10a^3b^2 \\pm 10a^2b^3 + 5ab^4 \\pm b^5 \\\\ \\ldots","$$ (a \\pm b)^4 = a^4 \\pm 4a^3b + 6a^2b^2 \\pm 4ab^3 + b^4 \\\\ (a \\pm b)^5 = a^5 \\pm 5a^4b + 10a^3b^2 \\pm 10a^2b^3 + 5ab^4 \\pm b^5 \\\\ \\ldots $$","blockMath-USBHSmZHI",{"__TSPROSE_proseElement":215,"schema":5434,"children":5435,"id":5439},{"name":229,"type":218,"linkable":215,"__TSPROSE_schema":215},[5436],{"__TSPROSE_proseElement":215,"schema":5437,"data":5438},{"name":233,"type":234,"linkable":219,"__TSPROSE_schema":215},"The Binomial Theorem shows up all over mathematics, both elementary and especially advanced. Just as these formulas let you turn sums into products of brackets and back again for small powers, the Binomial Theorem lets you do the same with expressions of any complexity.","paragraph-GQzH4upPK",{"$$ (x+5)^2 - (x-5)(x+5) - 10(x+5) $$":5441,"$$ 0 $$":5441,"$$ (x+5)^2 = (x+5)(x+5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25 $$":5441,"$$ (x-5)(x+5) = x^2 + \\cancel{5x} - \\cancel{5x} - 25 = x^2 - 25 $$":5441,"$ +5x $":5441,"$ -5x $":5441,"$$ 10(x+5) = 10x + 50 $$":5441,"$$ (x^2 + 10x + 25) - (x^2 - 25) - (10x + 50) = \\cancel{x^2} + \\cancel{10x} + 25 - \\cancel{x^2} + 25 - \\cancel{10x} - 50 = \\boxed{0} $$":5441,"$$ \\frac{(a+b)^2 - (a-b)^2}{4ab} $$":5441,"$$ 1 $$":5441,"$$ (a+b)^2 = (a + b)(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2 \\\\ (a-b)^2 = (a - b)(a-b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2 $$":5441,"$$ (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = \\cancel{a^2} + 2ab + \\cancel{b^2} - \\cancel{a^2} + 2ab - \\cancel{b^2} = 4ab $$":5441,"$ 4ab $":5441,"$$ \\frac{\\cancel{4ab}}{\\cancel{4ab}} = \\boxed{1} $$":5441,"$$ \\frac{(m+n)^2 - m^2 - n^2}{mn} \\cdot \\frac{(m-n)^2 - m^2 - n^2}{mn} $$":5441,"$$ -4 $$":5441,"$$ (m+n)^2 - m^2 - n^2 = \\cancel{m^2} + 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = 2mn $$":5441,"$$ (m-n)^2 - m^2 - n^2 = \\cancel{m^2} - 2mn + \\cancel{n^2} - \\cancel{m^2} - \\cancel{n^2} = -2mn $$":5441,"$$ \\frac{2\\cancel{mn}}{\\cancel{mn}} \\cdot \\frac{-2\\cancel{mn}}{\\cancel{mn}} = 2 \\cdot (-2) = \\boxed{-4} $$":5441,"$ 2 $":5441,"$ 90% $":5441,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fa-b-squared-meme.svg":5442,"$ (a+b)^n = a^n + b^n $":5441,"\u003Clink:external>\u002Fhttps:\u002F\u002Fw.wiki\u002FPjb":5446,"$$ \\red{(1+2)^2 = 1^2 + 2^2 = 5} \\\\ \\boxed{\\green{(1 + 2)^2 = 3^2 = 9}} >>{big} \\red{(2+3)^3 = 2^3 + 3^3 = 35} \\\\ \\boxed{\\green{(2 + 3)^3 = 5^3 = 125}} $$":5441,"$ (a+b)^2 $":5441,"$ a + b $":5441,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Ffoil.svg":5451,"$$ (a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \\boxed{a^2 + 2ab + b^2} $$":5441,"$ 2ab $":5441,"$$ a^2 + 2ab + b^2 = a^2 + ab + ab + b^2 = a(a + b) + b(a + b) = (a + b)(a + b) = \\boxed{(a + b)^2} $$":5441,"$ a $":5441,"$ b $":5441,"$ a^2 $":5441,"$ ab $":5441,"$ b^2 $":5441,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-sum-schema.svg":5455,"$$ (a + b)^2 = a^2 + 2ab + b^2 $$":5441,"$$ (m+5)^2 $$":5441,"$$ m^2 + 10m + 25 $$":5441,"$ m $":5441,"$ 5 $":5441,"$$ m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25 $$":5441,"$$ (m+5)^2 = m^2 + 10m + 25 $$":5441,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 $$":5441,"$$ 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":5441,"$ \\frac{1}{8}x $":5441,"$$ 2^2 = 4 >>{big} \\cancel{2} \\cdot \\cancel{2} \\cdot \\frac{1}{\\cancel{8}_{\\small\\cancel{4}_{\\small 2}}}x = \\frac{1}{2}x >>{big} \\left( \\frac{1}{8}x \\right)^2 = \\left(\\frac{1}{8}\\right)^2 x^2 = \\frac{1}{64}x^2 $$":5441,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":5441,"$$ (-7a - 3b)^2 $$":5441,"$ 1 $":5441,"$$ (-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2 $$":5441,"$$ 49a^2 + 42ab + 9b^2 $$":5441,"$ -7a - 3b $":5441,"$ -7a $":5441,"$ 3b $":5441,"$ -3b $":5441,"$$ (-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2 $$":5441,"$$ (-7a - 3b)^2 = 49a^2 + 42ab + 9b^2 $$":5441,"$$ -7a - 3b = -(7a + 3b) $$":5441,"$$ (-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2 $$":5441,"$$ 49 + 14x + x^2 $$":5441,"$$ (7 + x)^2 $$":5441,"$ 49 = 7^2 $":5441,"$ x^2 = x^2 $":5441,"$ a = 7 $":5441,"$ b = x $":5441,"$ 2ab = 2 \\cdot 7 \\cdot x = 14x $":5441,"$$ 49 + 14x + x^2 = \\underset{a^2}{7^2} + 2 \\cdot \\underset{a}{7} \\cdot \\underset{b}{x} + \\underset{b^2}{x^2} = (7 + x)^2 $$":5441,"$ 14x $":5441,"$ 7x $":5441,"$ 49 $":5441,"$ x^2 $":5441,"$ 7x = 7 \\cdot x $":5441,"$$ 1 + 8y + 16y^2 $$":5441,"$ 1^2 = 1 $":5441,"$$ (1 + 4y)^2 $$":5441,"$ 1 = 1^2 $":5441,"$ 16y^2 = (4y)^2 $":5441,"$ a = 1 $":5441,"$ b = 4y $":5441,"$ 2ab = 2 \\cdot 1 \\cdot 4y = 8y $":5441,"$$ 1 + 8y + 16y^2 = \\underset{a^2}{1^2} + 2 \\cdot \\underset{a}{1} \\cdot \\underset{b}{4y} + \\underset{b^2}{(4y)^2} = (1 + 4y)^2 $$":5441,"$ 8y $":5441,"$ 4y $":5441,"$ 16y^2 $":5441,"$ 1 \\cdot 4y $":5441,"$ 1^2 = 1 = a $":5441,"$ (4y)^2 = 16y^2 = b^2 $":5441,"$$ \\frac{1}{4}k^2 + k + 1 $$":5441,"$$ \\left(\\frac{k}{2} + 1 \\right)^2 $$":5441,"$ \\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2 $":5441,"$ a = \\frac{k}{2} $":5441,"$ b = 1 $":5441,"$ 2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k $":5441,"$$ \\frac{1}{4}k^2 + k + 1 = \\underset{a^2}{\\left( \\frac{k}{2} \\right)^2} + 2 \\cdot \\underset{a}{\\frac{k}{2}} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = \\left(\\frac{k}{2} + 1 \\right)^2 $$":5441,"$ k $":5441,"$ \\frac{k}{2} $":5441,"$ \\frac{1}{4}k^2 $":5441,"$ \\frac{k}{2} \\cdot 1 $":5441,"$ \\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2 $":5441,"$ 1^2 = 1 = b^2 $":5441,"$$ 16t^2 + 36m^2 + 48tm $$":5441,"$$ (4t + 6m)^2 $$":5441,"$$ 16t^2 + 48tm + 36m^2 $$":5441,"$ 16t^2 = (4t)^2 $":5441,"$ 36m^2 = (6m)^2 $":5441,"$ a = 4t $":5441,"$ b = 6m $":5441,"$ 2ab = 2 \\cdot 4t \\cdot 6m = 48tm $":5441,"$$ 16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2 $$":5441,"$ 48tm $":5441,"$ 24tm $":5441,"$ 16t^2 $":5441,"$ 36m^2 $":5441,"$ 24tm = 4t \\cdot 6m $":5441,"$ (4t)^2 = 16t^2 = a^2 $":5441,"$ (6m)^2 = 36m^2 = b^2 $":5441,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fancient-formula.webp":5459,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas":5461,"$ (a-b)^2 $":5441,"$ a-b $":5441,"$$ (a - b)^2 = (a - b)(a - b) = a^2 - ab - ba + b^2 = \\boxed{a^2 - 2ab + b^2} \\\\ a^2 - 2ab + b^2 = a^2 - ab - ab + b^2 = a(a - b) - b(a - b) = (a - b)(a - b) = \\boxed{(a - b)^2} $$":5441,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-diff-schema.svg":5469,"$$ (a - b)^2 = a^2 - 2ab + b^2 $$":5441,"$ 3-2 $":5441,"$ 3+(-2) $":5441,"$$ - a - b + c - d = (-a) + (-b) + c + (-d) $$":5441,"$ -2 $":5441,"$$ (6-c)^2 $$":5441,"$$ 36 - 12c + c^2 $$":5441,"$ 6 $":5441,"$ c $":5441,"$$ 6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2 $$":5441,"$$ (6-c)^2 = 36 - 12c + c^2 $$":5441,"$ -b $":5441,"$$ (6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2} $$":5441,"$$ 9x^2 - 6x + 1 $$":5441,"$$ (3x - 1)^2 $$":5441,"$ 9x^2 = (3x)^2 $":5441,"$ a = 3x $":5441,"$ -2ab = -2 \\cdot 3x \\cdot 1 = -6x $":5441,"$$ 9x^2 - 6x + 1 = \\underset{a^2}{(3x)^2} - 2 \\cdot \\underset{a}{3x} \\cdot \\underset{b}{1} + \\underset{b^2}{1^2} = (3x - 1)^2 $$":5441,"$ -6x $":5441,"$ 3x $":5441,"$ 9x^2 $":5441,"$ 3x = 3x \\cdot 1 $":5441,"$ 9x^2 = (3x)^2 = a^2 $":5441,"$ 1 = 1^2 = b^2 $":5441,"$$ (-7 + 2a)^2 $$":5441,"$$ 4a^2 - 28a + 49 $$":5441,"$$ (-7 + 2a)^2 = (2a - 7)^2 $$":5441,"$ 2a $":5441,"$ 7 $":5441,"$$ (2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49 $$":5441,"$$ (-7 + 2a)^2 = 4a^2 - 28a + 49 $$":5441,"$$ - 12k + 4k^2 + 9 $$":5441,"$$ (2k - 3)^2 $$":5441,"$$ - 12k + 4k^2 + 9 = 4k^2 - 12k + 9 $$":5441,"$ 4k^2 = (2k)^2 $":5441,"$ 9 = 3^2 $":5441,"$ a = 2k $":5441,"$ b = 3 $":5441,"$ -2ab = -2 \\cdot 2k \\cdot 3 = -12k $":5441,"$$ 4k^2 - 12k + 9 = \\underset{a^2}{(2k)^2} - 2 \\cdot \\underset{a}{2k} \\cdot \\underset{b}{3} + \\underset{b^2}{3^2} = (2k - 3)^2 $$":5441,"$ -12k $":5441,"$ 6k $":5441,"$ 4k^2 $":5441,"$ 9 $":5441,"$ 6k = 2k \\cdot 3 $":5441,"$ (2k)^2 = 4k^2 = a^2 $":5441,"$ 3^2 = 9 = b^2 $":5441,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 $$":5441,"$$ 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":5441,"$ 5y $":5441,"$ \\frac{3}{4}x $":5441,"$$ (5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2 $$":5441,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 = 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":5441,"$$ \\frac{1}{4}m^2 - 5m + 25 $$":5441,"$$ \\left(\\frac{m}{2} - 5\\right)^2 $$":5441,"$ \\frac{1}{4}m^2 = \\left(\\frac{m}{2}\\right)^2 $":5441,"$ 25 = 5^2 $":5441,"$ a = \\frac{m}{2} $":5441,"$ b = 5 $":5441,"$ -2ab = -2 \\cdot \\frac{m}{2} \\cdot 5 = -5m $":5441,"$$ \\frac{1}{4}m^2 - 5m + 25 = \\underset{a^2}{\\left(\\frac{1}{2}m\\right)^2} - 2 \\cdot \\underset{a}{\\frac{1}{2}m} \\cdot \\underset{b}{5} + \\underset{b^2}{5^2} = \\left(\\frac{m}{2} - 5\\right)^2 $$":5441,"$ -5m $":5441,"$ \\frac{5m}{2} $":5441,"$ \\frac{1}{4}m^2 $":5441,"$ 25 $":5441,"$ \\frac{5m}{2} = \\frac{m}{2} \\cdot 5 $":5441,"$ \\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2 $":5441,"$ 5^2 = 25 = b^2 $":5441,"$ 0 $":5441,"$ 10 $":5441,"$ 4^2 = 16 $":5441,"$ 6^2 = 36 $":5441,"$ 9^2 = 81 $":5441,"$ 20 $":5441,"$ 11^2 = 121 $":5441,"$ 15^2 = 225 $":5441,"$ 19^2 = 361 $":5441,"$ 99% $":5441,"$ > 15 $":5441,"$ 62 $":5441,"$ 60 $":5441,"$ 3844 $":5441,"$ 62 = 60 + 2 $":5441,"$ 62^2 $":5441,"$$ 62^2 = (60 + 2)^2 = 60^2 + 2 \\cdot 60 \\cdot 2 + 2^2 = 3600 + 240 + 4 = 3844 $$":5441,"$ 2 \\cdot 60 \\cdot 2 $":5441,"$ 3600 $":5441,"$ 240 $":5441,"$$ 24 \\Rightarrow 20 + 4 >>{big} 31 \\Rightarrow 30 + 1 >>{big} 44 \\Rightarrow 40 + 4 >>{big} 53 \\Rightarrow 50 + 3 $$":5441,"$ 48 $":5441,"$ 50 $":5441,"$ 2304 $":5441,"$ 48 = 50 - 2 $":5441,"$ 48^2 $":5441,"$$ 48^2 = (50 - 2)^2 = 50^2 - 2 \\cdot 50 \\cdot 2 + 2^2 = 2500 - 200 + 4 = 2304 $$":5441,"$ 2 \\cdot 50 \\cdot 2 $":5441,"$$ 27 \\Rightarrow 30 - 3 >>{big} 39 \\Rightarrow 40 - 1 >>{big} 46 \\Rightarrow 50 - 4 >>{big} 57 \\Rightarrow 60 - 3 $$":5441,"problemScript:content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fmental-squares":5473,"$ 42 $":5441,"$ 1764 $":5441,"$$ x^2 + 2x + 1 = 0 >>{big} 36 - 18t + 9t^2 = 0 >>{big} 4z^2 + 48z + 144 = 0 $$":5441,"$ 0 = 0 $":5441,"$$ (x+1)^2 = 0 >>{big} (6 - 3t)^2 = 0 >>{big} (2z + 12)^2 = 0 $$":5441,"$ x $":5441,"$ -1 $":5441,"$ t $":5441,"$ z $":5441,"$ 12 $":5441,"$ -6 $":5441,"$$ (\\underset{x}{-1} + 1)^2 = 0 >>{big} (6 - 3 \\cdot \\underset{t}{2})^2 = 0 >>{big} (2 \\cdot \\underset{z}{-6} + 12)^2 = 0 $$":5441,"\u003Clink:global>\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square":5475,"$ a^2 - b^2 $":5441,"$ a - b $":5441,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fdiff-of-squares.svg":5479,"$$ a^2 - b^2 = (a + b)(a - b) $$":5441,"$$ (a+b)(a-b) = a \\cdot a - a \\cdot b + b \\cdot a - b \\cdot b = \\boxed{a^2 - b^2} $$":5441,"$$ a^2 - b^2 = a^2 + \\underbrace{ab - ab}_{\\text{Added and subtracted}} - b^2 = a(a+b) - b(a+b) = \\boxed{(a+b)(a-b)} $$":5441,"$ (a - b)^2 $":5441,"$$ x^2 - 36 $$":5441,"$$ (x + 6)(x - 6) $$":5441,"$ x \\Rightarrow x^2 $":5441,"$ 6 \\Rightarrow 36 $":5441,"$$ x^2 - 36 = \\underset{a}{x^2} - \\underset{b}{6^2} = (x + 6)(x - 6) $$":5441,"$$ (3 - x)(3 + x) $$":5441,"$$ 9 - x^2 $$":5441,"$$ 3^2 = 9 >>{big} x^2 $$":5441,"$$ \\frac{4}{81}k^2 - \\frac{1}{25}b^2 $$":5441,"$$ \\left(\\frac{2k}{9} + \\frac{b}{5}\\right)\\left(\\frac{2k}{9} - \\frac{b}{5}\\right) $$":5441,"$$ \\frac{4}{81}k^2 = \\frac{2^2}{9^2}k^2 = \\left(\\frac{2k}{9}\\right)^2 >>{big} \\frac{1}{25}b^2 = \\frac{1^2}{5^2}b^2 = \\left(\\frac{b}{5}\\right)^2 $$":5441,"$ 2k\u002F9 $":5441,"$ b\u002F5 $":5441,"$$ \\frac{4}{81}k^2 - \\frac{1}{25}b^2 = \\underset{a}{\\left(\\frac{2k}{9}\\right)^2} - \\underset{b}{\\left(\\frac{b}{5}\\right)^2} = \\left(\\frac{2k}{9} + \\frac{b}{5}\\right)\\left(\\frac{2k}{9} - \\frac{b}{5}\\right) $$":5441,"$$ \\left( t + \\frac{3}{4} \\right)\\left( \\frac{3}{4} - t \\right) $$":5441,"$$ \\frac{9}{16} - t^2 $$":5441,"$$ \\left( t + \\frac{3}{4} \\right)\\left( \\frac{3}{4} - t \\right) = \\left( \\frac{3}{4} + t \\right)\\left( \\frac{3}{4} - t \\right) = \\ldots $$":5441,"$$ \\ldots = \\left( \\frac{3}{4} + t \\right)\\left( \\frac{3}{4} - t \\right) = \\left(\\frac{3}{4}\\right)^2 - t^2 = \\frac{9}{16} - t^2 $$":5441,"$$ 9x^4z^2 - 0.09y^2 $$":5441,"$$ (3x^2z + 0.3y)(3x^2z - 0.3y) $$":5441,"$$ 9x^4z^2 = 3^2(x^2)^2z^2 = (3x^2z)^2 >>{big} 0.09y^2 = \\frac{9y^2}{100} = \\frac{3^2y^2}{10^2} = \\left(\\frac{3y}{10}\\right)^2 = (0.3y)^2 $$":5441,"$ 3x^2z $":5441,"$ 0.3y $":5441,"$$ 9x^4z^2 - 0.09y^2 = \\underset{a}{(3x^2z)^2} - \\underset{b}{(0.3y)^2} = (3x^2z + 0.3y)(3x^2z - 0.3y) $$":5441,"$$ (a+b)^3 = (a+b)^2(a+b) = (a^2 + 2ab + b^2)(a+b) = \\boxed{a^3 + 3a^2b + 3ab^2 + b^3} \\\\ (a-b)^3 = (a-b)^2(a-b) = (a^2 - 2ab + b^2)(a-b) = \\boxed{a^3 - 3a^2b + 3ab^2 - b^3} $$":5441,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fpractice\u002F$cubeSumDiffFactorization":5483,"content\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fcube-sum.mp4":5491,"\u003Clink:external>\u002Fhttps:\u002F\u002Fwww.tiktok.com\u002F@complex_math\u002Fvideo\u002F7358570064724970759":5493,"$ 3 $":5441,"$ a^2b $":5441,"$ ab^2 $":5441,"$$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$":5441,"$$ (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $$":5441,"$$ (x + 2)^3 $$":5441,"$$ x^3 + 6x^2 + 12x + 8 $$":5441,"$$ (x + 2)^3 = x^3 + 3 \\cdot x^2 \\cdot 2 + 3 \\cdot x \\cdot 2^2 + 2^3 = x^3 + 6x^2 + 12x + 8 $$":5441,"$$ \\left(\\frac{1}{3} - \\frac{2}{k}\\right)^3 $$":5441,"$$ \\frac{1}{27} - \\frac{2}{3k} + \\frac{4}{k^2} - \\frac{8}{k^3} $$":5441,"$ 1\u002F3 $":5441,"$ 2\u002Fk $":5441,"$$ \\left(\\frac{1}{3} - \\frac{2}{k}\\right)^3 = \\left(\\frac{1}{3}\\right)^3 - 3\\left(\\frac{1}{3}\\right)^2 \\cdot \\frac{2}{k} + 3 \\cdot \\frac{1}{3} \\cdot \\left(\\frac{2}{k}\\right)^2 - \\left(\\frac{2}{k}\\right)^3 = \\\\ \\frac{1}{27} - \\cancel{3} \\cdot \\frac{1}{\\cancel{9}_{\\small 3}} \\cdot \\frac{2}{k} + \\cancel{3} \\cdot \\frac{1}{\\cancel{3}_{\\small 1}} \\cdot \\frac{4}{k^2} - \\frac{8}{k^3} = \\\\ \\frac{1}{27} - \\frac{2}{3k} + \\frac{4}{k^2} - \\frac{8}{k^3} $$":5441,"$$ m^3 + 6m^2n + 12mn^2 + 8n^3 $$":5441,"$$ (m + 2n)^3 $$":5441,"$ a^3 = m^3 $":5441,"$ a = m $":5441,"$ b^3 = 8n^3 = (2n)^3 $":5441,"$ b = 2n $":5441,"$$ \\underbrace{m^3}_{a^3} + 3 \\cdot \\underbrace{m^2}_{a^2} \\cdot \\underbrace{2n}_{b} + 3 \\cdot \\underbrace{m}_{a} \\cdot \\underbrace{(2n)^2}_{b^2} + \\underbrace{(2n)^3}_{b^3} = (m + 2n)^3 $$":5441,"$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} $$":5441,"$$ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$":5441,"$ a^3 = x^3\u002F8 = (x\u002F2)^3 $":5441,"$ a = x\u002F2 $":5441,"$ b^3 = y^3\u002F27 = (y\u002F3)^3 $":5441,"$ b = y\u002F3 $":5441,"$$ \\frac{x^3}{8} - \\frac{x^2y}{4} + \\frac{xy^2}{6} - \\frac{y^3}{27} = \\\\ \\underbrace{\\frac{x^3}{8}}_{a^3} - 3 \\cdot \\underbrace{\\frac{x^2}{4}}_{a^2} \\cdot \\underbrace{\\frac{y}{3}}_{b} + 3 \\cdot \\underbrace{\\frac{x}{2}}_{a} \\cdot \\underbrace{\\frac{y^2}{9}}_{b^2} - \\underbrace{\\frac{y^3}{27}}_{b^3} = \\\\ \\left(\\frac{x}{2} - \\frac{y}{3}\\right)^3 $$":5441,"$ (a + b)^2 $":5441,"$$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $$":5441,"$ (a+b)^2 = (a+b)(a+b) = \\ldots $":5441,"$ a^2-b^2 $":5441,"$ (a+b)(a-b) $":5441,"$$ (a \\pm b)^{\\normalsize\\brand{2}} = a^2 \\pm \\brand{2}ab + b^2 \\\\ (a \\pm b)^{\\normalsize\\brand{3}} = a^3 \\pm \\brand{3}a^2b + \\brand{3}ab^2 \\pm b^3 $$":5441,"$$ (a \\pm b)^1 = a \\pm b \\\\ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 \\\\ \\text{???} $$":5441,"$$ (a+b)^n = \\sum\\limits_{k=0}^{n} \\binom{n}{k} a^{n-k}b^k, \\quad \\text{where } \\binom{n}{k} = \\frac{n!}{k!(n-k)!} $$":5441,"$$ (a \\pm b)^4 = a^4 \\pm 4a^3b + 6a^2b^2 \\pm 4ab^3 + b^4 \\\\ (a \\pm b)^5 = a^5 \\pm 5a^4b + 10a^3b^2 \\pm 10a^2b^3 + 5ab^4 \\pm b^5 \\\\ \\ldots $$":5441},null,{"resolvedSrc":5443,"width":5444,"height":5445},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fa-b-squared-meme.svg",1079,1457,{"type":5447,"resolvedHref":5448,"previewRequest":5449},"external","https:\u002F\u002Fw.wiki\u002FPjb",{"type":5450,"href":5448},"direct-link",{"resolvedSrc":5452,"width":5453,"height":5454},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Ffoil.svg",1103,440,{"resolvedSrc":5456,"width":5457,"height":5458},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-sum-schema.svg",4182,1355,{"resolvedIconSrc":5460,"videoIcon":219},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fancient-formula.webp",{"type":5462,"content":5463,"resolvedHref":5465,"previewRequest":5466},"contentItem",{"contentType":181,"contentTitle":5464,"topicPart":210},"Vieta's Formulas","\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas\u002F",{"type":5467,"contentType":181,"topicPart":210,"fullId":5468},"content-page","foundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas",{"resolvedSrc":5470,"width":5471,"height":5472},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fsquare-diff-schema.svg",4363,1418,{"resolvedScriptSrc":5474},"\u002Fapi\u002FproblemScript\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fscripts\u002Fmental-squares.js",{"type":5462,"content":5476,"resolvedHref":183,"previewRequest":5477},{"contentType":181,"contentTitle":182,"topicPart":210},{"type":5467,"contentType":181,"topicPart":210,"fullId":5478},"foundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square",{"resolvedSrc":5480,"width":5481,"height":5482},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fdiff-of-squares.svg",5848,1384,{"type":5484,"schemaName":5485,"elementTitle":5486,"content":5487,"resolvedHref":5488,"previewRequest":5489},"unique","problem","Factoring the Cube of a Sum and a Difference",{"contentType":181,"contentTitle":160,"topicPart":213},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum-diff-factorization",{"type":5484,"contentFullId":158,"contentType":181,"topicPart":213,"uniqueName":5490},"cubeSumDiffFactorization",{"resolvedSrc":5492},"\u002Ffile\u002Fcontent\u002F01-foundations\u002F01-polynomials\u002F01-special-products\u002Fassets\u002Fcube-sum.mp4",{"type":5447,"resolvedHref":5494,"previewRequest":5495},"https:\u002F\u002Fwww.tiktok.com\u002F@complex_math\u002Fvideo\u002F7358570064724970759",{"type":5450,"href":5494},[5497,5502,5506,5510,5514,5519,5525,5530,5536,5542,5549,5557,5562,5567],{"link":5498,"schemaName":223,"title":225,"seo":5499,"description":5501},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=what-are-special-products",{"title":5500},"What are Special Products?","Special Products are formulas that let you quickly expand brackets or, going the other way, factor expressions back into brackets.",{"link":5498,"schemaName":533,"title":160,"key":5503,"seo":5504,"description":5505},{"title":5500},{},"Formulas that let you quickly \"unpack\" compact powered expressions into expansions or, going the other way,\n          \"pack\" long sums into a compact form. These formulas save you from doing routine calculations by hand.",{"link":188,"schemaName":189,"title":552,"key":5507,"seo":5508,"description":5509},{},{},"One of the Special Products formulas: (a+b)² = a² + 2ab + b².\n          It lets you quickly expand brackets or, the other way around, pack an expanded expression back into them.",{"link":193,"schemaName":189,"title":2110,"key":5511,"seo":5512,"description":5513},{},{},"One of the Special Products formulas: (a-b)² = a² - 2ab + b².\n          It lets you expand parentheses quickly or, going the other way, factor an expression back into squared parentheses.",{"link":5515,"schemaName":223,"title":3253,"seo":5516,"description":5518},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=applications-of-the-square-and-a-difference",{"title":5517},"Where are the square of a sum and the square of a difference used?","Practical examples of using the Special Products formulas for the square of a sum and the square of a difference in different areas of mathematics and real life.",{"link":5520,"schemaName":239,"title":3398,"key":5521,"seo":5522,"description":5524},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=fast-square",{},{"title":5523},"How can you square a number quickly in your head?","A universal method for quickly squaring any number by using the square-of-a-sum and square-of-a-difference formulas.\n          For numbers within 100, this process can often be done mentally.",{"link":5526,"schemaName":189,"title":4052,"key":5527,"seo":5528,"description":5529},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=diff-of-squares",{},{},"One of the Special Products formulas: a² - b² = (a + b)(a - b).\n          Lets you halve or double the degree.\n          Used to simplify expressions.",{"link":5531,"schemaName":189,"title":5532,"key":5533,"seo":5534,"description":5535},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum","Cube of a Sum",{},{},"One of the Special Products formulas: (a + b)³ = a³ + 3a²b + 3ab² + b³.\n          Lets you quickly expand the cube of a sum of two expressions.\n          Used to simplify expressions and solve equations.",{"link":5537,"schemaName":189,"title":5538,"key":5539,"seo":5540,"description":5541},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-diff","Cube of a Difference",{},{},"One of the Special Products formulas: (a - b)³ = a³ - 3a²b + 3ab² - b³.\n          Lets you quickly expand the cube of a difference of two expressions.\n          Used to simplify expressions and solve equations.",{"link":5543,"schemaName":5222,"title":5544,"key":5545,"seo":5546,"description":5548},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=memorization-tips","How to memorize?",{},{"title":5547},"How to memorize Special Products?","A set of tips and tricks that help you memorize Special Products and avoid mixing them up.",{"link":5550,"schemaName":5551,"title":5552,"key":5553,"seo":5555,"description":5556},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=special-products-table","table","Table of Special Products",{"title":5554},"Special Products table",{},"A single table with all the Special Products formulas.",{"link":5558,"schemaName":239,"title":5559,"seo":5560,"description":5561},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=practice-square-of-a-sum-and-square-of-a-difference","Problems on the square of a sum and the square of a difference",{},"A large set of problems for practicing the Special Products formulas for the square of a sum and the square of a difference.\n          Two types of problems: expand and rewrite as the square of a sum or a difference.",{"link":5563,"schemaName":239,"title":5564,"seo":5565,"description":5566},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=practice-difference-of-squares","Problems on the difference of squares",{},"A large set of problems for practicing the difference of squares formula.\n          Two types of problems: expand and rewrite as a difference of squares.",{"link":5568,"schemaName":239,"title":5569,"seo":5570,"description":5571},"\u002Fpractice\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=practice-cube-of-a-sum-and-cube-of-a-difference","Problems on the cube of a sum and the cube of a difference",{},"A large set of problems for practicing the Special Products formulas for the cube of a sum and the cube of a difference.\n          Two types of problems: expand and rewrite as the cube of a sum or a 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Equations",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Felementary\u002F",[210,212,213],{"type":8,"shortId":5621,"title":5622,"flags":5623,"link":5624},"foundations\u002Fequations\u002Fzero-product-property","Zero Product Property",{},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fzero-product-property\u002F",{"type":5600,"separator":219,"shortId":5626,"title":5627,"flags":5628,"link":5629,"children":5630},"foundations\u002Fequations\u002Fquadratic","Quadratic Equations",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002Fquadratic\u002F",[5631,5637,5643,5646,5652,5658,5663,5666,5672],{"type":181,"shortId":5632,"title":5633,"flags":5634,"link":5635,"parts":5636},"foundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it","What is it?",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it\u002F",[210,212,213],{"type":181,"shortId":5638,"title":5639,"flags":5640,"link":5641,"parts":5642},"foundations\u002Fequations\u002Fquadratic\u002Fincomplete","With missing terms",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fincomplete\u002F",[210,212,213],{"type":181,"shortId":5478,"title":182,"flags":5644,"link":183,"parts":5645},{},[210,212,213],{"type":181,"shortId":5647,"title":5648,"flags":5649,"link":5650,"parts":5651},"foundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula","Quadratic formula",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002F",[210,212,213],{"type":181,"shortId":5653,"title":5654,"flags":5655,"link":5656,"parts":5657},"foundations\u002Fequations\u002Fquadratic\u002Ffactoring","Factoring quadratics",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002F",[210,212,213],{"type":8,"shortId":5659,"title":5660,"flags":5661,"link":5662},"foundations\u002Fequations\u002Fquadratic\u002Freal-world","Real Life",{"secondary":215},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F",{"type":181,"shortId":5468,"title":5464,"flags":5664,"link":5465,"parts":5665},{"secondary":215},[210,212,213],{"type":181,"shortId":5667,"title":5668,"flags":5669,"link":5670,"parts":5671},"foundations\u002Fequations\u002Fquadratic\u002Fmental-solving","Mental solving",{"secondary":215,"advanced":215},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002F",[210,212],{"type":181,"shortId":5673,"title":5674,"flags":5675,"link":5676,"parts":5677},"foundations\u002Fequations\u002Fquadratic\u002Fformulas","General 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