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Do not forget to reduce fractions.","paragraph-AGv7U4x7E",{"__TSPROSE_proseElement":209,"schema":559,"data":560,"storageKey":562,"id":563},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":561,"freeze":213},"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","$$ 2^2 = 4 >>{big} \\cancel{2} \\cdot \\cancel{2} \\cdot \\frac{1}{\\cancel{8}_{\\small\\cancel{4}_{\\small 2}}}x = \\frac{1}{2}x >>{big} \\left( \\frac{1}{8}x \\right)^2 = \\left(\\frac{1}{8}\\right)^2 x^2 = \\frac{1}{64}x^2 $$","blockMath-UEXl8k1FG",{"__TSPROSE_proseElement":209,"schema":565,"children":566,"id":569},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[567],{"__TSPROSE_proseElement":209,"schema":568,"data":485},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-MaZtcGLZc-1",{"__TSPROSE_proseElement":209,"schema":571,"data":572,"storageKey":574,"id":575},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":573,"freeze":213},"\\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2","$$ \\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$","blockMath-iW3hsY3J6",{"__TSPROSE_proseElement":209,"schema":577,"data":578,"children":579},{"name":396,"type":212,"linkable":213,"__TSPROSE_schema":209},{},[580,589,595,628,636],{"__TSPROSE_proseElement":209,"schema":581,"children":582},{"name":401,"type":212,"linkable":213,"__TSPROSE_schema":209},[583],{"__TSPROSE_proseElement":209,"schema":584,"data":585,"storageKey":587,"id":588},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":586,"freeze":213},"(-7a - 3b)^2","$$ (-7a - 3b)^2 $$","blockMath-FV48nxGyl",{"__TSPROSE_proseElement":209,"schema":590,"data":591},{"name":412,"type":212,"linkable":213,"__TSPROSE_schema":209},{"serializedValidator":592},{"__ERUDIT_CHECK":209,"name":415,"data":593},{"expr":594},"49a^2 + 42ab + 9b^2",{"__TSPROSE_proseElement":209,"schema":596,"children":598},{"name":597,"type":212,"linkable":213,"__TSPROSE_schema":209},"problemHint",[599,615,621],{"__TSPROSE_proseElement":209,"schema":600,"children":601,"id":614},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[602,605,611],{"__TSPROSE_proseElement":209,"schema":603,"data":604},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Option ",{"__TSPROSE_proseElement":209,"schema":606,"data":607,"storageKey":609,"id":610},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":608},"1","$ 1 $","inlinerMath-GWJ3eIxvK",{"__TSPROSE_proseElement":209,"schema":612,"data":613},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," - rewrite the expression under the square as a sum with an explicit plus sign:","paragraph-IgTve8X1e",{"__TSPROSE_proseElement":209,"schema":616,"data":617,"storageKey":619,"id":620},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":618,"freeze":213},"(-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2","$$ (-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2 $$","blockMath-YyTXKOh9m",{"__TSPROSE_proseElement":209,"schema":622,"children":623,"id":627},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[624],{"__TSPROSE_proseElement":209,"schema":625,"data":626},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Option 2 -- factor a minus sign out inside the sum.","paragraph-mut7KohdA",{"__TSPROSE_proseElement":209,"schema":629,"children":630},{"name":420,"type":212,"linkable":213,"__TSPROSE_schema":209},[631],{"__TSPROSE_proseElement":209,"schema":632,"data":633,"storageKey":634,"id":635},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":594,"freeze":213},"$$ 49a^2 + 42ab + 9b^2 $$","blockMath-JHMiXhAI7",{"__TSPROSE_proseElement":209,"schema":637,"children":638},{"name":429,"type":212,"linkable":213,"__TSPROSE_schema":209},[639,754],{"__TSPROSE_proseElement":209,"schema":640,"data":642,"children":643},{"name":641,"type":212,"linkable":213,"__TSPROSE_schema":209},"problemSection","Using an explicit plus sign",[644,700,704,736,742,748],{"__TSPROSE_proseElement":209,"schema":645,"children":646,"id":699},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[647,650,656,659,665,668,674,677,681,684,688,690,696],{"__TSPROSE_proseElement":209,"schema":648,"data":649},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"The minus sign in ",{"__TSPROSE_proseElement":209,"schema":651,"data":652,"storageKey":654,"id":655},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":653},"-7a - 3b","$ -7a - 3b $","inlinerMath-EOtl20nRc",{"__TSPROSE_proseElement":209,"schema":657,"data":658},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," can be interpreted in two ways. On one hand, you can read it as the difference of ",{"__TSPROSE_proseElement":209,"schema":660,"data":661,"storageKey":663,"id":664},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":662},"-7a","$ -7a $","inlinerMath-80LHk8XHo",{"__TSPROSE_proseElement":209,"schema":666,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," and ",{"__TSPROSE_proseElement":209,"schema":669,"data":670,"storageKey":672,"id":673},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":671},"3b","$ 3b $","inlinerMath-OkRX6uD9W",{"__TSPROSE_proseElement":209,"schema":675,"data":676},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". 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Then you get a sum of two numbers, ",{"__TSPROSE_proseElement":209,"schema":685,"data":686,"storageKey":663,"id":687},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":662},"inlinerMath-80LHk8XHo-1",{"__TSPROSE_proseElement":209,"schema":689,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":691,"data":692,"storageKey":694,"id":695},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":693},"-3b","$ -3b $","inlinerMath-Djaufz4Sl",{"__TSPROSE_proseElement":209,"schema":697,"data":698},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},", which can be written explicitly like this:","paragraph-U25VE6X3m",{"__TSPROSE_proseElement":209,"schema":701,"data":702,"storageKey":619,"id":703},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":618,"freeze":213},"blockMath-YyTXKOh9m-1",{"__TSPROSE_proseElement":209,"schema":705,"children":706,"id":735},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[707,710,714,716,720,722,726,728,732],{"__TSPROSE_proseElement":209,"schema":708,"data":709},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"After that, it is the same procedure as before. In the square of a sum formula, the role of ",{"__TSPROSE_proseElement":209,"schema":711,"data":712,"storageKey":441,"id":713},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":440},"inlinerMath-YqDfaaRXA-2",{"__TSPROSE_proseElement":209,"schema":715,"data":445},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":717,"data":718,"storageKey":663,"id":719},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":662},"inlinerMath-80LHk8XHo-2",{"__TSPROSE_proseElement":209,"schema":721,"data":454},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":723,"data":724,"storageKey":459,"id":725},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":458},"inlinerMath-LTS8nNQCZ-2",{"__TSPROSE_proseElement":209,"schema":727,"data":463},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":729,"data":730,"storageKey":694,"id":731},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":693},"inlinerMath-Djaufz4Sl-1",{"__TSPROSE_proseElement":209,"schema":733,"data":734},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". 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":209,"schema":737,"data":738,"storageKey":740,"id":741},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":739,"freeze":213},"(-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":209,"schema":743,"children":744,"id":747},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[745],{"__TSPROSE_proseElement":209,"schema":746,"data":485},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-MaZtcGLZc-2",{"__TSPROSE_proseElement":209,"schema":749,"data":750,"storageKey":752,"id":753},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":751,"freeze":213},"(-7a - 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That gives ",{"__TSPROSE_proseElement":209,"schema":1278,"data":1279,"storageKey":1281,"id":1282},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1280},"\\frac{k}{2}","$ \\frac{k}{2} $","inlinerMath-IcB1YEh3y",{"__TSPROSE_proseElement":209,"schema":1284,"data":1285},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". Now write that fraction as a product of two factors whose squares match the outer terms ",{"__TSPROSE_proseElement":209,"schema":1287,"data":1288,"storageKey":1290,"id":1291},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1289},"\\frac{1}{4}k^2","$ \\frac{1}{4}k^2 $","inlinerMath-XhsYjlucP",{"__TSPROSE_proseElement":209,"schema":1293,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1295,"data":1296,"storageKey":609,"id":1297},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":608},"inlinerMath-GWJ3eIxvK-2",{"__TSPROSE_proseElement":209,"schema":1299,"data":1120},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1301,"data":1302,"storageKey":1006,"id":1303},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1005},"inlinerMath-ooHpGun5c-2",{"__TSPROSE_proseElement":209,"schema":1305,"data":1127},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1307,"data":1308,"storageKey":1310,"id":1311},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1309},"\\frac{k}{2} \\cdot 1","$ \\frac{k}{2} \\cdot 1 $","inlinerMath-5VcE9HQud",{"__TSPROSE_proseElement":209,"schema":1313,"data":1136},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1315,"data":1316,"storageKey":1318,"id":1319},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1317},"\\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2","$ \\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2 $","inlinerMath-9J027juAN",{"__TSPROSE_proseElement":209,"schema":1321,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1323,"data":1324,"storageKey":1326,"id":1327},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1325},"1^2 = 1 = b^2","$ 1^2 = 1 = b^2 $","inlinerMath-5XaGNX6Ix",{"__TSPROSE_proseElement":209,"schema":1329,"data":1153},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-N5QsfrYkL",{"__TSPROSE_proseElement":209,"schema":1332,"data":1333,"storageKey":1252,"id":1334},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1251,"freeze":213},"blockMath-OAtFOiXrE-1",{"__TSPROSE_proseElement":209,"schema":1336,"data":1337,"children":1338},{"name":396,"type":212,"linkable":213,"__TSPROSE_schema":209},{},[1339,1348,1354,1364,1372],{"__TSPROSE_proseElement":209,"schema":1340,"children":1341},{"name":401,"type":212,"linkable":213,"__TSPROSE_schema":209},[1342],{"__TSPROSE_proseElement":209,"schema":1343,"data":1344,"storageKey":1346,"id":1347},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1345,"freeze":213},"16t^2 + 36m^2 + 48tm","$$ 16t^2 + 36m^2 + 48tm $$","blockMath-BRO83bY8l",{"__TSPROSE_proseElement":209,"schema":1349,"data":1350},{"name":412,"type":212,"linkable":213,"__TSPROSE_schema":209},{"serializedValidator":1351},{"__ERUDIT_CHECK":209,"name":415,"data":1352},{"expr":1353},"(4t + 6m)^2",{"__TSPROSE_proseElement":209,"schema":1355,"children":1356},{"name":597,"type":212,"linkable":213,"__TSPROSE_schema":209},[1357],{"__TSPROSE_proseElement":209,"schema":1358,"children":1359,"id":1363},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1360],{"__TSPROSE_proseElement":209,"schema":1361,"data":1362},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"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":209,"schema":1365,"children":1366},{"name":420,"type":212,"linkable":213,"__TSPROSE_schema":209},[1367],{"__TSPROSE_proseElement":209,"schema":1368,"data":1369,"storageKey":1370,"id":1371},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1353,"freeze":213},"$$ (4t + 6m)^2 $$","blockMath-YexRM6ll2",{"__TSPROSE_proseElement":209,"schema":1373,"children":1374},{"name":429,"type":212,"linkable":213,"__TSPROSE_schema":209},[1375,1382,1388,1443],{"__TSPROSE_proseElement":209,"schema":1376,"children":1377,"id":1381},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1378],{"__TSPROSE_proseElement":209,"schema":1379,"data":1380},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"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":209,"schema":1383,"data":1384,"storageKey":1386,"id":1387},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1385,"freeze":213},"16t^2 + 48tm + 36m^2","$$ 16t^2 + 48tm + 36m^2 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(6m)^2","$ 36m^2 = (6m)^2 $","inlinerMath-44R58sl9K",{"__TSPROSE_proseElement":209,"schema":1411,"data":861},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1413,"data":1414,"storageKey":1416,"id":1417},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1415},"a = 4t","$ a = 4t $","inlinerMath-Dvo0ZB2yR",{"__TSPROSE_proseElement":209,"schema":1419,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1421,"data":1422,"storageKey":1424,"id":1425},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1423},"b = 6m","$ b = 6m $","inlinerMath-zfqINXAo1",{"__TSPROSE_proseElement":209,"schema":1427,"data":878},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1429,"data":1430,"storageKey":1432,"id":1433},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1431},"2ab = 2 \\cdot 4t \\cdot 6m = 48tm","$ 2ab = 2 \\cdot 4t \\cdot 6m = 48tm $","inlinerMath-ukNlMgNXk",{"__TSPROSE_proseElement":209,"schema":1435,"data":1246},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-EANb24h54",{"__TSPROSE_proseElement":209,"schema":1438,"data":1439,"storageKey":1441,"id":1442},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1440,"freeze":213},"16t^2 + 36m^2 + 48tm = \\underset{a^2}{(4t)^2} + 2 \\cdot \\underset{a}{4t} \\cdot \\underset{b}{6m} + \\underset{b^2}{(6m)^2} = (4t + 6m)^2","$$ 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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$","inlinerMath-CGznEo7ze",{"__TSPROSE_proseElement":209,"schema":1516,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1518,"data":1519,"storageKey":1521,"id":1522},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1520},"(6m)^2 = 36m^2 = b^2","$ (6m)^2 = 36m^2 = b^2 $","inlinerMath-VR8AvPckN",{"__TSPROSE_proseElement":209,"schema":1524,"data":1153},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-jRC2NPmU1",{"__TSPROSE_proseElement":209,"schema":1527,"data":1528,"storageKey":1441,"id":1529},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1440,"freeze":213},"blockMath-akHjm2BVw-1","packing-into-the-square-of-a-sum",{"__TSPROSE_proseElement":209,"schema":1532,"data":1533,"children":1536,"id":2468},{"name":380,"type":212,"linkable":209,"__TSPROSE_schema":209},{"title":1534,"level":383,"attributes":1535},"Examples Using the Square of a Difference",[],[1537,1673,1879,1993,2195,2286],{"__TSPROSE_proseElement":209,"schema":1538,"data":1539,"children":1540},{"name":396,"type":212,"linkable":213,"__TSPROSE_schema":209},{},[1541,1557,1563,1571,1632],{"__TSPROSE_proseElement":209,"schema":1542,"children":1543},{"name":401,"type":212,"linkable":213,"__TSPROSE_schema":209},[1544,1551],{"__TSPROSE_proseElement":209,"schema":1545,"children":1546,"id":1550},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1547],{"__TSPROSE_proseElement":209,"schema":1548,"data":1549},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Expand the parentheses:","paragraph-qYJVaoVtK",{"__TSPROSE_proseElement":209,"schema":1552,"data":1553,"storageKey":1555,"id":1556},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1554,"freeze":213},"(6-c)^2","$$ (6-c)^2 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$","inlinerMath-9rSWhNw25",{"__TSPROSE_proseElement":209,"schema":1594,"data":1595},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},", and ",{"__TSPROSE_proseElement":209,"schema":1597,"data":1598,"storageKey":459,"id":1599},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":458},"inlinerMath-LTS8nNQCZ-3",{"__TSPROSE_proseElement":209,"schema":1601,"data":1602},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," by the letter ",{"__TSPROSE_proseElement":209,"schema":1604,"data":1605,"storageKey":1607,"id":1608},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1606},"c","$ c $","inlinerMath-8Adf0APnu",{"__TSPROSE_proseElement":209,"schema":1610,"data":1611},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". From left to right, find the square of the first term, the negative of twice the product of the first and second terms, and the square of the second term. With numbers this simple, it is best to do the computation mentally:","paragraph-YaZv649zg",{"__TSPROSE_proseElement":209,"schema":1614,"data":1615,"storageKey":1617,"id":1618},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1616,"freeze":213},"6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2","$$ 6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2 $$","blockMath-goZZUGvb0",{"__TSPROSE_proseElement":209,"schema":1620,"children":1621,"id":1625},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1622],{"__TSPROSE_proseElement":209,"schema":1623,"data":1624},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Write all the results together:","paragraph-xv2qdT6ML",{"__TSPROSE_proseElement":209,"schema":1627,"data":1628,"storageKey":1630,"id":1631},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1629,"freeze":213},"(6-c)^2 = 36 - 12c + c^2","$$ (6-c)^2 = 36 - 12c + c^2 $$","blockMath-1J8WJP6EN",{"__TSPROSE_proseElement":209,"schema":1633,"children":1635},{"name":1634,"type":212,"linkable":213,"__TSPROSE_schema":209},"problemNote",[1636,1652,1658],{"__TSPROSE_proseElement":209,"schema":1637,"children":1638,"id":1651},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1639,1642,1648],{"__TSPROSE_proseElement":209,"schema":1640,"data":1641},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Many beginners often make the double minus mistake. When applying the square-of-a-difference formula, they take one minus sign from the formula and a second one from ",{"__TSPROSE_proseElement":209,"schema":1643,"data":1644,"storageKey":1646,"id":1647},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1645},"-b","$ -b $","inlinerMath-MMRLAwV3s",{"__TSPROSE_proseElement":209,"schema":1649,"data":1650},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". As a result, minus times minus becomes plus, and they get the wrong answer:","paragraph-NxEixv6Kh",{"__TSPROSE_proseElement":209,"schema":1653,"data":1654,"storageKey":1656,"id":1657},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1655,"freeze":213},"(6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2}","$$ (6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2} $$","blockMath-ucfrketpu",{"__TSPROSE_proseElement":209,"schema":1659,"children":1660,"id":1672},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1661,1664],{"__TSPROSE_proseElement":209,"schema":1662,"data":1663},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Remember once and for all: ",{"__TSPROSE_proseElement":209,"schema":1665,"data":1666,"children":1667,"id":1671},{"name":232,"type":222,"linkable":209,"__TSPROSE_schema":209},{"type":234},[1668],{"__TSPROSE_proseElement":209,"schema":1669,"data":1670},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"any minus sign has already been accounted for in the square of a difference. Do not think about signs separately at all!","emphasis-scv1WyRre","paragraph-316s7d4GN",{"__TSPROSE_proseElement":209,"schema":1674,"data":1675,"children":1676},{"name":396,"type":212,"linkable":213,"__TSPROSE_schema":209},{},[1677,1693,1699,1716,1724],{"__TSPROSE_proseElement":209,"schema":1678,"children":1679},{"name":401,"type":212,"linkable":213,"__TSPROSE_schema":209},[1680,1687],{"__TSPROSE_proseElement":209,"schema":1681,"children":1682,"id":1686},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1683],{"__TSPROSE_proseElement":209,"schema":1684,"data":1685},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Write the sum as a square of a difference:","paragraph-MudZgQrHU",{"__TSPROSE_proseElement":209,"schema":1688,"data":1689,"storageKey":1691,"id":1692},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1690,"freeze":213},"9x^2 - 6x + 1","$$ 9x^2 - 6x + 1 $$","blockMath-1DwYljmox",{"__TSPROSE_proseElement":209,"schema":1694,"data":1695},{"name":412,"type":212,"linkable":213,"__TSPROSE_schema":209},{"serializedValidator":1696},{"__ERUDIT_CHECK":209,"name":415,"data":1697},{"expr":1698},"(3x - 1)^2",{"__TSPROSE_proseElement":209,"schema":1700,"children":1701},{"name":597,"type":212,"linkable":213,"__TSPROSE_schema":209},[1702],{"__TSPROSE_proseElement":209,"schema":1703,"children":1704,"id":1715},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1705,1708,1712],{"__TSPROSE_proseElement":209,"schema":1706,"data":1707},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"You have not forgotten that ",{"__TSPROSE_proseElement":209,"schema":1709,"data":1710,"storageKey":1006,"id":1711},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1005},"inlinerMath-ooHpGun5c-3",{"__TSPROSE_proseElement":209,"schema":1713,"data":1714},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},", have you?","paragraph-Prk67niCV",{"__TSPROSE_proseElement":209,"schema":1717,"children":1718},{"name":420,"type":212,"linkable":213,"__TSPROSE_schema":209},[1719],{"__TSPROSE_proseElement":209,"schema":1720,"data":1721,"storageKey":1722,"id":1723},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1698,"freeze":213},"$$ (3x - 1)^2 $$","blockMath-p7CiNdIap",{"__TSPROSE_proseElement":209,"schema":1725,"children":1726},{"name":429,"type":212,"linkable":213,"__TSPROSE_schema":209},[1727,1781],{"__TSPROSE_proseElement":209,"schema":1728,"data":1729,"children":1730},{"name":641,"type":212,"linkable":213,"__TSPROSE_schema":209},"Using the Outer Terms",[1731,1775],{"__TSPROSE_proseElement":209,"schema":1732,"children":1733,"id":1774},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1734,1736,1742,1744,1748,1751,1757,1759,1763,1765,1771],{"__TSPROSE_proseElement":209,"schema":1735,"data":844},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1737,"data":1738,"storageKey":1740,"id":1741},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1739},"9x^2 = (3x)^2","$ 9x^2 = (3x)^2 $","inlinerMath-JHiC7V8Qr",{"__TSPROSE_proseElement":209,"schema":1743,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1745,"data":1746,"storageKey":1032,"id":1747},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1031},"inlinerMath-AM9HQ4pvg-2",{"__TSPROSE_proseElement":209,"schema":1749,"data":1750},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". Both are nice squares, so we can see right away that ",{"__TSPROSE_proseElement":209,"schema":1752,"data":1753,"storageKey":1755,"id":1756},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1754},"a = 3x","$ a = 3x $","inlinerMath-gOFjcUN5E",{"__TSPROSE_proseElement":209,"schema":1758,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1760,"data":1761,"storageKey":1234,"id":1762},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1233},"inlinerMath-jT8WRtcC5-1",{"__TSPROSE_proseElement":209,"schema":1764,"data":878},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1766,"data":1767,"storageKey":1769,"id":1770},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1768},"-2ab = -2 \\cdot 3x \\cdot 1 = -6x","$ -2ab = -2 \\cdot 3x \\cdot 1 = -6x $","inlinerMath-bd0EntIKb",{"__TSPROSE_proseElement":209,"schema":1772,"data":1773},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," and it matches. So we write:","paragraph-4CV7UyT1a",{"__TSPROSE_proseElement":209,"schema":1776,"data":1777,"storageKey":1779,"id":1780},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1778,"freeze":213},"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":209,"schema":1782,"data":1783,"children":1784},{"name":641,"type":212,"linkable":213,"__TSPROSE_schema":209},"Using the Middle Term",[1785,1875],{"__TSPROSE_proseElement":209,"schema":1786,"children":1787,"id":1874},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1788,1790,1796,1798,1804,1806,1812,1815,1821,1823,1827,1830,1834,1837,1843,1845,1849,1851,1855,1857,1863,1865,1871],{"__TSPROSE_proseElement":209,"schema":1789,"data":904},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1791,"data":1792,"storageKey":1794,"id":1795},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1793},"-6x","$ -6x $","inlinerMath-MG3Leuh7I",{"__TSPROSE_proseElement":209,"schema":1797,"data":913},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1799,"data":1800,"storageKey":1802,"id":1803},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1801},"-2","$ -2 $","inlinerMath-02bjIi1Ry",{"__TSPROSE_proseElement":209,"schema":1805,"data":920},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1807,"data":1808,"storageKey":1810,"id":1811},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1809},"3x","$ 3x $","inlinerMath-WYRLesvjE",{"__TSPROSE_proseElement":209,"schema":1813,"data":1814},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". This product must be split into two factors whose squares match the outer terms ",{"__TSPROSE_proseElement":209,"schema":1816,"data":1817,"storageKey":1819,"id":1820},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1818},"9x^2","$ 9x^2 $","inlinerMath-oeT1YYBtQ",{"__TSPROSE_proseElement":209,"schema":1822,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1824,"data":1825,"storageKey":609,"id":1826},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":608},"inlinerMath-GWJ3eIxvK-3",{"__TSPROSE_proseElement":209,"schema":1828,"data":1829},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". Since ",{"__TSPROSE_proseElement":209,"schema":1831,"data":1832,"storageKey":1006,"id":1833},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1005},"inlinerMath-ooHpGun5c-4",{"__TSPROSE_proseElement":209,"schema":1835,"data":1836},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},", we artificially include a factor of one: ",{"__TSPROSE_proseElement":209,"schema":1838,"data":1839,"storageKey":1841,"id":1842},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1840},"3x = 3x \\cdot 1","$ 3x = 3x \\cdot 1 $","inlinerMath-nbioH3US9",{"__TSPROSE_proseElement":209,"schema":1844,"data":955},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1846,"data":1847,"storageKey":1755,"id":1848},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1754},"inlinerMath-gOFjcUN5E-1",{"__TSPROSE_proseElement":209,"schema":1850,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1852,"data":1853,"storageKey":1234,"id":1854},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1233},"inlinerMath-jT8WRtcC5-2",{"__TSPROSE_proseElement":209,"schema":1856,"data":1136},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1858,"data":1859,"storageKey":1861,"id":1862},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1860},"9x^2 = (3x)^2 = a^2","$ 9x^2 = (3x)^2 = a^2 $","inlinerMath-BPY4xTos9",{"__TSPROSE_proseElement":209,"schema":1864,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1866,"data":1867,"storageKey":1869,"id":1870},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1868},"1 = 1^2 = b^2","$ 1 = 1^2 = b^2 $","inlinerMath-BRasBzsqn",{"__TSPROSE_proseElement":209,"schema":1872,"data":1873},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". Everything matches, so we write the square of a difference:","paragraph-2M1PIaNFI",{"__TSPROSE_proseElement":209,"schema":1876,"data":1877,"storageKey":1779,"id":1878},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1778,"freeze":213},"blockMath-y8LnumBRS-1",{"__TSPROSE_proseElement":209,"schema":1880,"data":1881,"children":1882},{"name":396,"type":212,"linkable":213,"__TSPROSE_schema":209},{},[1883,1898,1904,1914,1922],{"__TSPROSE_proseElement":209,"schema":1884,"children":1885},{"name":401,"type":212,"linkable":213,"__TSPROSE_schema":209},[1886,1892],{"__TSPROSE_proseElement":209,"schema":1887,"children":1888,"id":1891},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1889],{"__TSPROSE_proseElement":209,"schema":1890,"data":1549},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-qYJVaoVtK-1",{"__TSPROSE_proseElement":209,"schema":1893,"data":1894,"storageKey":1896,"id":1897},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1895,"freeze":213},"(-7 + 2a)^2","$$ (-7 + 2a)^2 $$","blockMath-UAZ3MmPrY",{"__TSPROSE_proseElement":209,"schema":1899,"data":1900},{"name":412,"type":212,"linkable":213,"__TSPROSE_schema":209},{"serializedValidator":1901},{"__ERUDIT_CHECK":209,"name":415,"data":1902},{"expr":1903},"4a^2 - 28a + 49",{"__TSPROSE_proseElement":209,"schema":1905,"children":1906},{"name":597,"type":212,"linkable":213,"__TSPROSE_schema":209},[1907],{"__TSPROSE_proseElement":209,"schema":1908,"children":1909,"id":1913},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1910],{"__TSPROSE_proseElement":209,"schema":1911,"data":1912},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Use the fact that changing the order of addends does not change a sum.","paragraph-eYPBu4RpC",{"__TSPROSE_proseElement":209,"schema":1915,"children":1916},{"name":420,"type":212,"linkable":213,"__TSPROSE_schema":209},[1917],{"__TSPROSE_proseElement":209,"schema":1918,"data":1919,"storageKey":1920,"id":1921},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1903,"freeze":213},"$$ 4a^2 - 28a + 49 $$","blockMath-UKdlYItNi",{"__TSPROSE_proseElement":209,"schema":1923,"children":1924},{"name":429,"type":212,"linkable":213,"__TSPROSE_schema":209},[1925,1932,1938,1975,1981,1987],{"__TSPROSE_proseElement":209,"schema":1926,"children":1927,"id":1931},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1928],{"__TSPROSE_proseElement":209,"schema":1929,"data":1930},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"This sum has an unusual form. The minus sign is not in the middle but at the beginning. Swap the first and second terms (the sum does not change) so it is easier to compare with the square-of-a-difference formula:","paragraph-yU7hAKPve",{"__TSPROSE_proseElement":209,"schema":1933,"data":1934,"storageKey":1936,"id":1937},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1935,"freeze":213},"(-7 + 2a)^2 = (2a - 7)^2","$$ (-7 + 2a)^2 = (2a - 7)^2 $$","blockMath-vPY8ftQWb",{"__TSPROSE_proseElement":209,"schema":1939,"children":1940,"id":1974},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1941,1944,1948,1951,1957,1959,1963,1965,1971],{"__TSPROSE_proseElement":209,"schema":1942,"data":1943},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Now ",{"__TSPROSE_proseElement":209,"schema":1945,"data":1946,"storageKey":441,"id":1947},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":440},"inlinerMath-YqDfaaRXA-4",{"__TSPROSE_proseElement":209,"schema":1949,"data":1950},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," in the formula is played by ",{"__TSPROSE_proseElement":209,"schema":1952,"data":1953,"storageKey":1955,"id":1956},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1954},"2a","$ 2a $","inlinerMath-uofr24ZDY",{"__TSPROSE_proseElement":209,"schema":1958,"data":1595},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1960,"data":1961,"storageKey":459,"id":1962},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":458},"inlinerMath-LTS8nNQCZ-4",{"__TSPROSE_proseElement":209,"schema":1964,"data":463},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":1966,"data":1967,"storageKey":1969,"id":1970},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1968},"7","$ 7 $","inlinerMath-sYh3uEvdQ",{"__TSPROSE_proseElement":209,"schema":1972,"data":1973},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". 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":209,"schema":1976,"data":1977,"storageKey":1979,"id":1980},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1978,"freeze":213},"(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 $$","blockMath-iIPFwePt6",{"__TSPROSE_proseElement":209,"schema":1982,"children":1983,"id":1986},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[1984],{"__TSPROSE_proseElement":209,"schema":1985,"data":1624},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-xv2qdT6ML-1",{"__TSPROSE_proseElement":209,"schema":1988,"data":1989,"storageKey":1991,"id":1992},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":1990,"freeze":213},"(-7 + 2a)^2 = 4a^2 - 28a + 49","$$ (-7 + 2a)^2 = 4a^2 - 28a + 49 $$","blockMath-CQRD6g7ty",{"__TSPROSE_proseElement":209,"schema":1994,"data":1995,"children":1996},{"name":396,"type":212,"linkable":213,"__TSPROSE_schema":209},{},[1997,2012,2018,2028,2036],{"__TSPROSE_proseElement":209,"schema":1998,"children":1999},{"name":401,"type":212,"linkable":213,"__TSPROSE_schema":209},[2000,2006],{"__TSPROSE_proseElement":209,"schema":2001,"children":2002,"id":2005},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2003],{"__TSPROSE_proseElement":209,"schema":2004,"data":1685},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-MudZgQrHU-1",{"__TSPROSE_proseElement":209,"schema":2007,"data":2008,"storageKey":2010,"id":2011},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":2009,"freeze":213},"- 12k + 4k^2 + 9","$$ - 12k + 4k^2 + 9 $$","blockMath-6wHlw0P5m",{"__TSPROSE_proseElement":209,"schema":2013,"data":2014},{"name":412,"type":212,"linkable":213,"__TSPROSE_schema":209},{"serializedValidator":2015},{"__ERUDIT_CHECK":209,"name":415,"data":2016},{"expr":2017},"(2k - 3)^2",{"__TSPROSE_proseElement":209,"schema":2019,"children":2020},{"name":597,"type":212,"linkable":213,"__TSPROSE_schema":209},[2021],{"__TSPROSE_proseElement":209,"schema":2022,"children":2023,"id":2027},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2024],{"__TSPROSE_proseElement":209,"schema":2025,"data":2026},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Reorder the terms so they match the order in the square-of-a-difference formula.","paragraph-uMbxwUl3s",{"__TSPROSE_proseElement":209,"schema":2029,"children":2030},{"name":420,"type":212,"linkable":213,"__TSPROSE_schema":209},[2031],{"__TSPROSE_proseElement":209,"schema":2032,"data":2033,"storageKey":2034,"id":2035},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":2017,"freeze":213},"$$ (2k - 3)^2 $$","blockMath-pruShJQyn",{"__TSPROSE_proseElement":209,"schema":2037,"children":2038},{"name":429,"type":212,"linkable":213,"__TSPROSE_schema":209},[2039,2046,2052,2107],{"__TSPROSE_proseElement":209,"schema":2040,"children":2041,"id":2045},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2042],{"__TSPROSE_proseElement":209,"schema":2043,"data":2044},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"This sum has an unusual form: the pure squares are not at the ends. 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From left to right, find the square of the first term, the negative of twice the product of the first and second terms, and the square of the second term. The numbers are more complicated here, so it is reasonable to compute all three separately on paper. Do not forget to simplify.","paragraph-UU93cM1FU",{"__TSPROSE_proseElement":209,"schema":2269,"data":2270,"storageKey":2272,"id":2273},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":2271,"freeze":213},"(5y)^2 = 5^2y^2 = 25y^2 >>{big} -\\cancel{2} \\cdot 5y \\cdot \\frac{3}{\\cancel{4}_{\\small 2}}x = -\\frac{15}{2}xy >>{big} \\left( \\frac{3}{4}x \\right)^2 = \\left(\\frac{3}{4}\\right)^2 x^2 = \\frac{9}{16}x^2","$$ (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 $$","blockMath-ZNKwF9cMD",{"__TSPROSE_proseElement":209,"schema":2275,"children":2276,"id":2279},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2277],{"__TSPROSE_proseElement":209,"schema":2278,"data":1624},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-xv2qdT6ML-2",{"__TSPROSE_proseElement":209,"schema":2281,"data":2282,"storageKey":2284,"id":2285},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":2283,"freeze":213},"\\left( 5y - 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5m + 25 = \\underset{a^2}{\\left(\\frac{1}{2}m\\right)^2} - 2 \\cdot \\underset{a}{\\frac{1}{2}m} \\cdot \\underset{b}{5} + \\underset{b^2}{5^2} = \\left(\\frac{m}{2} - 5\\right)^2","$$ \\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 $$","blockMath-xBGmZf40t",{"__TSPROSE_proseElement":209,"schema":2379,"data":1783,"children":2380},{"name":641,"type":212,"linkable":213,"__TSPROSE_schema":209},[2381,2464],{"__TSPROSE_proseElement":209,"schema":2382,"children":2383,"id":2463},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2384,2386,2392,2394,2398,2401,2407,2410,2416,2418,2424,2427,2433,2435,2439,2441,2445,2447,2453,2455,2461],{"__TSPROSE_proseElement":209,"schema":2385,"data":904},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2387,"data":2388,"storageKey":2390,"id":2391},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2389},"-5m","$ -5m $","inlinerMath-Aa3jYyDxo",{"__TSPROSE_proseElement":209,"schema":2393,"data":913},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2395,"data":2396,"storageKey":1802,"id":2397},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":1801},"inlinerMath-02bjIi1Ry-2",{"__TSPROSE_proseElement":209,"schema":2399,"data":2400},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},", even though it does not divide evenly. That gives ",{"__TSPROSE_proseElement":209,"schema":2402,"data":2403,"storageKey":2405,"id":2406},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2404},"\\frac{5m}{2}","$ \\frac{5m}{2} $","inlinerMath-l3IYVzMkX",{"__TSPROSE_proseElement":209,"schema":2408,"data":2409},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". Now write this fraction as a product of two factors whose squares match the outer terms ",{"__TSPROSE_proseElement":209,"schema":2411,"data":2412,"storageKey":2414,"id":2415},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2413},"\\frac{1}{4}m^2","$ \\frac{1}{4}m^2 $","inlinerMath-xhICl1aoZ",{"__TSPROSE_proseElement":209,"schema":2417,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2419,"data":2420,"storageKey":2422,"id":2423},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2421},"25","$ 25 $","inlinerMath-iCjF04534",{"__TSPROSE_proseElement":209,"schema":2425,"data":2426},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},". We factor it as ",{"__TSPROSE_proseElement":209,"schema":2428,"data":2429,"storageKey":2431,"id":2432},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2430},"\\frac{5m}{2} = \\frac{m}{2} \\cdot 5","$ \\frac{5m}{2} = \\frac{m}{2} \\cdot 5 $","inlinerMath-qax14IJhI",{"__TSPROSE_proseElement":209,"schema":2434,"data":955},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2436,"data":2437,"storageKey":2351,"id":2438},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2350},"inlinerMath-eUTZMMfdg-1",{"__TSPROSE_proseElement":209,"schema":2440,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2442,"data":2443,"storageKey":2359,"id":2444},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2358},"inlinerMath-pB2WH7ceV-1",{"__TSPROSE_proseElement":209,"schema":2446,"data":1136},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2448,"data":2449,"storageKey":2451,"id":2452},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2450},"\\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":209,"schema":2454,"data":667},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2456,"data":2457,"storageKey":2459,"id":2460},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2458},"5^2 = 25 = b^2","$ 5^2 = 25 = b^2 $","inlinerMath-tDEurJ8Oh",{"__TSPROSE_proseElement":209,"schema":2462,"data":1873},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-99UAlHyil",{"__TSPROSE_proseElement":209,"schema":2465,"data":2466,"storageKey":2376,"id":2467},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":2375,"freeze":213},"blockMath-xBGmZf40t-1","examples-using-the-square-of-a-difference","square-sum-diff-details",{"__TSPROSE_proseElement":209,"schema":2471},{"name":2472,"type":212,"linkable":213,"__TSPROSE_schema":209},"hr",{"__TSPROSE_proseElement":209,"schema":2474,"data":2476,"children":2478,"id":2490},{"name":2475,"type":212,"linkable":209,"__TSPROSE_schema":209},"accent_term",{"title":154,"layout":2477},"column",[2479],{"__TSPROSE_proseElement":209,"schema":2480,"children":2482},{"name":2481,"type":212,"linkable":213,"__TSPROSE_schema":209},"accentMain_term",[2483],{"__TSPROSE_proseElement":209,"schema":2484,"children":2485,"id":2489},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2486],{"__TSPROSE_proseElement":209,"schema":2487,"data":2488},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"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","special-products",{"__TSPROSE_proseElement":209,"schema":2492},{"name":2472,"type":212,"linkable":213,"__TSPROSE_schema":209},{"__TSPROSE_proseElement":209,"schema":2494,"data":2496,"children":2498,"id":2647},{"name":2495,"type":212,"linkable":209,"__TSPROSE_schema":209},"list",{"type":2497},"unordered",[2499,2535,2597],{"__TSPROSE_proseElement":209,"schema":2500,"children":2502},{"name":2501,"type":212,"linkable":213,"__TSPROSE_schema":209},"listItem",[2503,2515,2522,2528],{"__TSPROSE_proseElement":209,"schema":2504,"children":2505,"id":2514},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2506],{"__TSPROSE_proseElement":209,"schema":2507,"data":2508,"children":2509,"id":2513},{"name":232,"type":222,"linkable":209,"__TSPROSE_schema":209},{"type":234},[2510],{"__TSPROSE_proseElement":209,"schema":2511,"data":2512},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"Plus-minus sign","emphasis-1NyqRxQyB","paragraph-04bbQkxbo",{"__TSPROSE_proseElement":209,"schema":2516,"children":2517,"id":2521},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2518],{"__TSPROSE_proseElement":209,"schema":2519,"data":2520},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"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":209,"schema":2523,"data":2524,"storageKey":2526,"id":2527},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":2525,"freeze":213},"(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":209,"schema":2529,"children":2530,"id":2534},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2531],{"__TSPROSE_proseElement":209,"schema":2532,"data":2533},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"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":209,"schema":2536,"children":2537},{"name":2501,"type":212,"linkable":213,"__TSPROSE_schema":209},[2538,2550,2563],{"__TSPROSE_proseElement":209,"schema":2539,"children":2540,"id":2549},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2541],{"__TSPROSE_proseElement":209,"schema":2542,"data":2543,"children":2544,"id":2548},{"name":232,"type":222,"linkable":209,"__TSPROSE_schema":209},{"type":234},[2545],{"__TSPROSE_proseElement":209,"schema":2546,"data":2547},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"The difference of squares stands apart","emphasis-kyRJimTRg","paragraph-EKll4URRU",{"__TSPROSE_proseElement":209,"schema":2551,"children":2552,"id":2562},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2553,2556],{"__TSPROSE_proseElement":209,"schema":2554,"data":2555},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"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":209,"schema":2557,"data":2558,"storageKey":2560,"id":2561},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2559},"(a+b)^2 = (a+b)(a+b) = \\ldots","$ (a+b)^2 = (a+b)(a+b) = \\ldots $","inlinerMath-E2hB8EXr8","paragraph-xTUIHDIgz",{"__TSPROSE_proseElement":209,"schema":2564,"children":2565,"id":2596},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2566,2569,2575,2578,2584,2587,2593],{"__TSPROSE_proseElement":209,"schema":2567,"data":2568},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"But the difference of squares ",{"__TSPROSE_proseElement":209,"schema":2570,"data":2571,"storageKey":2573,"id":2574},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2572},"a^2-b^2","$ a^2-b^2 $","inlinerMath-Z18OlR0cJ",{"__TSPROSE_proseElement":209,"schema":2576,"data":2577},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," stands apart. First, it factors into brackets with a plus and a minus. Second, from the form ",{"__TSPROSE_proseElement":209,"schema":2579,"data":2580,"storageKey":2582,"id":2583},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2581},"a^2 - b^2","$ a^2 - b^2 $","inlinerMath-T1YOXnC6X",{"__TSPROSE_proseElement":209,"schema":2585,"data":2586},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," you cannot naturally and explicitly get the product ",{"__TSPROSE_proseElement":209,"schema":2588,"data":2589,"storageKey":2591,"id":2592},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2590},"(a+b)(a-b)","$ (a+b)(a-b) $","inlinerMath-bZ4wF1qyF",{"__TSPROSE_proseElement":209,"schema":2594,"data":2595},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209}," from the difference itself.","paragraph-bJ4EZXAyz",{"__TSPROSE_proseElement":209,"schema":2598,"children":2599},{"name":2501,"type":212,"linkable":213,"__TSPROSE_schema":209},[2600,2612,2634,2640],{"__TSPROSE_proseElement":209,"schema":2601,"children":2602,"id":2611},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2603],{"__TSPROSE_proseElement":209,"schema":2604,"data":2605,"children":2606,"id":2610},{"name":232,"type":222,"linkable":209,"__TSPROSE_schema":209},{"type":234},[2607],{"__TSPROSE_proseElement":209,"schema":2608,"data":2609},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"The degree matches the coefficient","emphasis-3bQgNnAYx","paragraph-SOnNSs7tQ",{"__TSPROSE_proseElement":209,"schema":2613,"children":2614,"id":2633},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2615,2618,2622,2625,2631],{"__TSPROSE_proseElement":209,"schema":2616,"data":2617},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"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":209,"schema":2619,"data":2620,"storageKey":538,"id":2621},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":537},"inlinerMath-z9Yar9waf-5",{"__TSPROSE_proseElement":209,"schema":2623,"data":2624},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},", and for the cube it is ",{"__TSPROSE_proseElement":209,"schema":2626,"data":2627,"storageKey":2629,"id":2630},{"name":276,"type":222,"linkable":209,"__TSPROSE_schema":209},{"katex":2628},"3","$ 3 $","inlinerMath-mqn34wvB0",{"__TSPROSE_proseElement":209,"schema":2632,"data":770},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"paragraph-rDkGDBMEF",{"__TSPROSE_proseElement":209,"schema":2635,"data":2636,"storageKey":2638,"id":2639},{"name":405,"type":212,"linkable":209,"__TSPROSE_schema":209},{"katex":2637,"freeze":213},"(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 $$","blockMath-L9UAp9IUH",{"__TSPROSE_proseElement":209,"schema":2641,"children":2642,"id":2646},{"name":388,"type":212,"linkable":209,"__TSPROSE_schema":209},[2643],{"__TSPROSE_proseElement":209,"schema":2644,"data":2645},{"name":238,"type":222,"linkable":213,"__TSPROSE_schema":209},"The coefficient is also easy to remember from the geometric derivation. For the square formulas, we build a square, and in the process two rectangles appear. For the cube formulas, we build a cube, and in the process two kinds of three parallelepipeds appear.","paragraph-JZ241jGeh","how-to-memorize",{"$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 $":2649,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Fsummary\u002F$squareSumDiffDetails":2650,"$ a^2 - b^2 = (a+b)(a-b) $":2649,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$diffOfSquaresExamples":2656,"$ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $":2649,"\u003Clink:global>\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002Farticle\u002F$cubeSumDiffExamples":2662,"$$ (m+5)^2 $$":2649,"$$ m^2 + 10m + 25 $$":2649,"$ a $":2649,"$ m $":2649,"$ b $":2649,"$ 5 $":2649,"$$ m^2 >>{big} 2 \\cdot m \\cdot 5 = 10m >>{big} 5^2 = 25 $$":2649,"$$ (m+5)^2 = m^2 + 10m + 25 $$":2649,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 $$":2649,"$$ 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":2649,"$ 2 $":2649,"$ \\frac{1}{8}x $":2649,"$$ 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 $$":2649,"$$ \\left( 2 + \\frac{1}{8}x \\right)^2 = 4 + \\frac{1}{2}x + \\frac{1}{64}x^2 $$":2649,"$$ (-7a - 3b)^2 $$":2649,"$ 1 $":2649,"$$ (-7a - 3b)^2 = \\left( (-7a) + (-3b) \\right)^2 $$":2649,"$$ 49a^2 + 42ab + 9b^2 $$":2649,"$ -7a - 3b $":2649,"$ -7a $":2649,"$ 3b $":2649,"$ -3b $":2649,"$$ (-7a)^2 = (-7)^2 a^2 = 49a^2 >>{big} 2 \\cdot (-7a) \\cdot (-3b) = 42ab >>{big} (-3b)^2 = (-3)^2 b^2 = 9b^2 $$":2649,"$$ (-7a - 3b)^2 = 49a^2 + 42ab + 9b^2 $$":2649,"$$ -7a - 3b = -(7a + 3b) $$":2649,"$$ (-(7a + 3b))^2 = (-1)^2 \\cdot (7a + 3b)^2 = (7a + 3b)^2 = 49a^2 + 42ab + 9b^2 $$":2649,"$$ 49 + 14x + x^2 $$":2649,"$$ (7 + x)^2 $$":2649,"$ 49 = 7^2 $":2649,"$ x^2 = x^2 $":2649,"$ a = 7 $":2649,"$ b = x $":2649,"$ 2ab = 2 \\cdot 7 \\cdot x = 14x $":2649,"$$ 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 $$":2649,"$ 14x $":2649,"$ 7x $":2649,"$ 49 $":2649,"$ x^2 $":2649,"$ 7x = 7 \\cdot x $":2649,"$$ 1 + 8y + 16y^2 $$":2649,"$ 1^2 = 1 $":2649,"$$ (1 + 4y)^2 $$":2649,"$ 1 = 1^2 $":2649,"$ 16y^2 = (4y)^2 $":2649,"$ a = 1 $":2649,"$ b = 4y $":2649,"$ 2ab = 2 \\cdot 1 \\cdot 4y = 8y $":2649,"$$ 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 $$":2649,"$ 8y $":2649,"$ 4y $":2649,"$ 16y^2 $":2649,"$ 1 \\cdot 4y $":2649,"$ 1^2 = 1 = a $":2649,"$ (4y)^2 = 16y^2 = b^2 $":2649,"$$ \\frac{1}{4}k^2 + k + 1 $$":2649,"$$ \\left(\\frac{k}{2} + 1 \\right)^2 $$":2649,"$ \\frac{1}{4}k^2 = \\left(\\frac{k}{2}\\right)^2 $":2649,"$ a = \\frac{k}{2} $":2649,"$ b = 1 $":2649,"$ 2ab = 2 \\cdot \\frac{k}{2} \\cdot 1 = k $":2649,"$$ \\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 $$":2649,"$ k $":2649,"$ \\frac{k}{2} $":2649,"$ \\frac{1}{4}k^2 $":2649,"$ \\frac{k}{2} \\cdot 1 $":2649,"$ \\left( \\frac{k}{2} \\right)^2 = \\frac{1}{4}k^2 = a^2 $":2649,"$ 1^2 = 1 = b^2 $":2649,"$$ 16t^2 + 36m^2 + 48tm $$":2649,"$$ (4t + 6m)^2 $$":2649,"$$ 16t^2 + 48tm + 36m^2 $$":2649,"$ 16t^2 = (4t)^2 $":2649,"$ 36m^2 = (6m)^2 $":2649,"$ a = 4t $":2649,"$ b = 6m $":2649,"$ 2ab = 2 \\cdot 4t \\cdot 6m = 48tm $":2649,"$$ 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 $$":2649,"$ 48tm $":2649,"$ 24tm $":2649,"$ 16t^2 $":2649,"$ 36m^2 $":2649,"$ 24tm = 4t \\cdot 6m $":2649,"$ (4t)^2 = 16t^2 = a^2 $":2649,"$ (6m)^2 = 36m^2 = b^2 $":2649,"$$ (6-c)^2 $$":2649,"$$ 36 - 12c + c^2 $$":2649,"$ 6 $":2649,"$ c $":2649,"$$ 6^2 = 36 >>{big} -2 \\cdot 6 \\cdot c = -12c >>{big} c^2 $$":2649,"$$ (6-c)^2 = 36 - 12c + c^2 $$":2649,"$ -b $":2649,"$$ (6-c)^2 = 36 - 2 \\cdot 6 \\cdot (-c) + c^2 = \\red{36 + 12c + c^2} $$":2649,"$$ 9x^2 - 6x + 1 $$":2649,"$$ (3x - 1)^2 $$":2649,"$ 9x^2 = (3x)^2 $":2649,"$ a = 3x $":2649,"$ -2ab = -2 \\cdot 3x \\cdot 1 = -6x $":2649,"$$ 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 $$":2649,"$ -6x $":2649,"$ -2 $":2649,"$ 3x $":2649,"$ 9x^2 $":2649,"$ 3x = 3x \\cdot 1 $":2649,"$ 9x^2 = (3x)^2 = a^2 $":2649,"$ 1 = 1^2 = b^2 $":2649,"$$ (-7 + 2a)^2 $$":2649,"$$ 4a^2 - 28a + 49 $$":2649,"$$ (-7 + 2a)^2 = (2a - 7)^2 $$":2649,"$ 2a $":2649,"$ 7 $":2649,"$$ (2a)^2 = 4a^2 >>{big} -2 \\cdot 2a \\cdot 7 = -28a >>{big} 7^2 = 49 $$":2649,"$$ (-7 + 2a)^2 = 4a^2 - 28a + 49 $$":2649,"$$ - 12k + 4k^2 + 9 $$":2649,"$$ (2k - 3)^2 $$":2649,"$$ - 12k + 4k^2 + 9 = 4k^2 - 12k + 9 $$":2649,"$ 4k^2 = (2k)^2 $":2649,"$ 9 = 3^2 $":2649,"$ a = 2k $":2649,"$ b = 3 $":2649,"$ -2ab = -2 \\cdot 2k \\cdot 3 = -12k $":2649,"$$ 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 $$":2649,"$ -12k $":2649,"$ 6k $":2649,"$ 4k^2 $":2649,"$ 9 $":2649,"$ 6k = 2k \\cdot 3 $":2649,"$ (2k)^2 = 4k^2 = a^2 $":2649,"$ 3^2 = 9 = b^2 $":2649,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 $$":2649,"$$ 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":2649,"$ 5y $":2649,"$ \\frac{3}{4}x $":2649,"$$ (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 $$":2649,"$$ \\left( 5y - \\frac{3}{4}x \\right)^2 = 25y^2 - \\frac{15}{2}xy + \\frac{9}{16}x^2 $$":2649,"$$ \\frac{1}{4}m^2 - 5m + 25 $$":2649,"$$ \\left(\\frac{m}{2} - 5\\right)^2 $$":2649,"$ \\frac{1}{4}m^2 = \\left(\\frac{m}{2}\\right)^2 $":2649,"$ 25 = 5^2 $":2649,"$ a = \\frac{m}{2} $":2649,"$ b = 5 $":2649,"$ -2ab = -2 \\cdot \\frac{m}{2} \\cdot 5 = -5m $":2649,"$$ \\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 $$":2649,"$ -5m $":2649,"$ \\frac{5m}{2} $":2649,"$ \\frac{1}{4}m^2 $":2649,"$ 25 $":2649,"$ \\frac{5m}{2} = \\frac{m}{2} \\cdot 5 $":2649,"$ \\left(\\frac{m}{2}\\right)^2 = \\frac{1}{4}m^2 = a^2 $":2649,"$ 5^2 = 25 = b^2 $":2649,"$$ (a \\pm b)^2 = a^2 \\pm 2ab + b^2 \\\\ (a \\pm b)^3 = a^3 \\pm 3a^2b + 3ab^2 \\pm b^3 $$":2649,"$ (a+b)^2 = (a+b)(a+b) = \\ldots $":2649,"$ a^2-b^2 $":2649,"$ a^2 - b^2 $":2649,"$ (a+b)(a-b) $":2649,"$ 3 $":2649,"$$ (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 $$":2649},null,{"type":2651,"schemaName":373,"elementTitle":376,"content":2652,"resolvedHref":2653,"previewRequest":2654},"unique",{"contentType":175,"contentTitle":154,"topicPart":204},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=square-sum-diff-details",{"type":2651,"contentFullId":152,"contentType":175,"topicPart":204,"uniqueName":2655},"squareSumDiffDetails",{"type":2651,"schemaName":380,"elementTitle":2657,"content":2658,"resolvedHref":2659,"previewRequest":2660},"Examples Using the Difference of Squares",{"contentType":175,"contentTitle":154,"topicPart":206},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=diff-of-squares-examples",{"type":2651,"contentFullId":152,"contentType":175,"topicPart":206,"uniqueName":2661},"diffOfSquaresExamples",{"type":2651,"schemaName":380,"elementTitle":2663,"content":2664,"resolvedHref":2665,"previewRequest":2666},"Examples Using the Cube of a Sum and Cube of a Difference",{"contentType":175,"contentTitle":154,"topicPart":206},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=cube-sum-diff-examples",{"type":2651,"contentFullId":152,"contentType":175,"topicPart":206,"uniqueName":2667},"cubeSumDiffExamples",[2669,2676,2680,2685,2690,2696,2703,2709,2715,2721,2728,2735,2740,2745],{"link":2670,"schemaName":2671,"title":2672,"seo":2673,"description":2675},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=what-are-special-products","heading","What Are Special Products?",{"title":2674},"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":2670,"schemaName":2475,"title":154,"key":2677,"seo":2678,"description":2679},{"title":2674},{},"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":182,"schemaName":183,"title":2681,"key":2682,"seo":2683,"description":2684},"Square of a Sum",{},{},"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":187,"schemaName":183,"title":2686,"key":2687,"seo":2688,"description":2689},"Square of a Difference",{},{},"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":2691,"schemaName":2671,"title":2692,"seo":2693,"description":2695},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=applications-of-the-square-and-a-difference","Applications of the Square and a Difference",{"title":2694},"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":2697,"schemaName":380,"title":2698,"key":2699,"seo":2700,"description":2702},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=fast-square","Quick Squares",{},{"title":2701},"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":2704,"schemaName":183,"title":2705,"key":2706,"seo":2707,"description":2708},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=diff-of-squares","Difference 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":2710,"schemaName":183,"title":2711,"key":2712,"seo":2713,"description":2714},"\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":2716,"schemaName":183,"title":2717,"key":2718,"seo":2719,"description":2720},"\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":2722,"schemaName":2495,"title":2723,"key":2724,"seo":2725,"description":2727},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=memorization-tips","How to memorize?",{},{"title":2726},"How to memorize Special Products?","A set of tips and tricks that help you memorize Special Products and avoid mixing them up.",{"link":2729,"schemaName":217,"title":2730,"key":2731,"seo":2733,"description":2734},"\u002Fsummary\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F?element=special-products-table","Table of Special Products",{"title":2732},"Special Products table",{},"A single table with all the Special Products formulas.",{"link":2736,"schemaName":380,"title":2737,"seo":2738,"description":2739},"\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":2741,"schemaName":380,"title":2742,"seo":2743,"description":2744},"\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":2746,"schemaName":380,"title":2747,"seo":2748,"description":2749},"\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 difference.",[2751,2753,2754],{"type":2752,"schemaName":217,"title":2732,"elementId":370},"element",{"type":2752,"schemaName":2475,"title":2674,"elementId":2490},{"type":2752,"schemaName":2495,"title":2723,"elementId":2647},{"contributorsCount":167,"sponsorsCount":2756},0,{"shortBookId":5,"frontNav":2758},{"type":14,"shortId":5,"title":15,"flags":2759,"link":17,"children":2760},{},[2761,2770],{"type":2762,"separator":209,"shortId":2763,"title":160,"flags":2764,"link":161,"children":2765},"group","foundations\u002Fpolynomials",{},[2766],{"type":175,"shortId":152,"title":154,"flags":2767,"link":2768,"parts":2769},{},"\u002Farticle\u002Ffoundations\u002Fpolynomials\u002Fspecial-products\u002F",[206,204,207],{"type":2762,"separator":209,"shortId":2771,"title":2772,"flags":2773,"link":2774,"children":2775},"foundations\u002Fequations","Equations",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002F",[2776,2782,2787,2844],{"type":175,"shortId":2777,"title":2778,"flags":2779,"link":2780,"parts":2781},"foundations\u002Fequations\u002Felementary","Elementary Equations",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Felementary\u002F",[206,204,207],{"type":8,"shortId":2783,"title":2784,"flags":2785,"link":2786},"foundations\u002Fequations\u002Fzero-product-property","Zero Product Property",{},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fzero-product-property\u002F",{"type":2762,"separator":213,"shortId":2788,"title":2789,"flags":2790,"link":2791,"children":2792},"foundations\u002Fequations\u002Fquadratic","Quadratic Equations",{},"\u002Fgroup\u002Ffoundations\u002Fequations\u002Fquadratic\u002F",[2793,2799,2805,2809,2815,2821,2826,2832,2838],{"type":175,"shortId":2794,"title":2795,"flags":2796,"link":2797,"parts":2798},"foundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it","What is it?",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fwhat-is-it\u002F",[206,204,207],{"type":175,"shortId":2800,"title":2801,"flags":2802,"link":2803,"parts":2804},"foundations\u002Fequations\u002Fquadratic\u002Fincomplete","With missing terms",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fincomplete\u002F",[206,204,207],{"type":175,"shortId":2806,"title":176,"flags":2807,"link":177,"parts":2808},"foundations\u002Fequations\u002Fquadratic\u002Fcompleting-the-square",{},[206,204,207],{"type":175,"shortId":2810,"title":2811,"flags":2812,"link":2813,"parts":2814},"foundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula","Quadratic formula",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fquadratic-formula\u002F",[206,204,207],{"type":175,"shortId":2816,"title":2817,"flags":2818,"link":2819,"parts":2820},"foundations\u002Fequations\u002Fquadratic\u002Ffactoring","Factoring quadratics",{},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Ffactoring\u002F",[206,204,207],{"type":8,"shortId":2822,"title":2823,"flags":2824,"link":2825},"foundations\u002Fequations\u002Fquadratic\u002Freal-world","Real Life",{"secondary":209},"\u002Fpage\u002Ffoundations\u002Fequations\u002Fquadratic\u002Freal-world\u002F",{"type":175,"shortId":2827,"title":2828,"flags":2829,"link":2830,"parts":2831},"foundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas","Vieta's Formulas",{"secondary":209},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fvietas-formulas\u002F",[206,204,207],{"type":175,"shortId":2833,"title":2834,"flags":2835,"link":2836,"parts":2837},"foundations\u002Fequations\u002Fquadratic\u002Fmental-solving","Mental solving",{"secondary":209,"advanced":209},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fmental-solving\u002F",[206,204],{"type":175,"shortId":2839,"title":2840,"flags":2841,"link":2842,"parts":2843},"foundations\u002Fequations\u002Fquadratic\u002Fformulas","General formulas",{"secondary":209},"\u002Fsummary\u002Ffoundations\u002Fequations\u002Fquadratic\u002Fformulas\u002F",[204,207],{"type":175,"shortId":2845,"title":2846,"flags":2847,"link":2848,"parts":2849},"foundations\u002Fequations\u002Fequivalence","Equivalence of equations",{"dev":209},"\u002Farticle\u002Ffoundations\u002Fequations\u002Fequivalence\u002F",[206],{"cameo":2851},[2852,2853,2854,2855,2856,2857,2858,2859,2860],"cameo-black-hole","cameo-confucius","cameo-dunno","cameo-einstein","cameo-euclid","cameo-gagarin","cameo-gauss","cameo-newton","cameo-t800",1777652714983]