Math FoundationsEquationsQuadratic Equations

Completing the Square

The technique of “packing” an expanded expression into a compact squared binomial. This method allows you to solve any quadratic equation! Let's learn how to do it with detailed examples and visualizations.
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Important1
Problem11
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Summary
Practice

Perfect square

😀
Elementary
Write as a perfect square:
x2+2xy+y2x^2 + 2xy + y^2

Complete square with a remainder

😀
Elementary
Complete the square:
1)m2−24m1)\enspace m^2 - 24m
2)x2−11x 2)\enspace x^2 - 11x
3)t2+5t−6 3)\enspace t^2 + 5t - 6
4)9x2−5x+7 4)\enspace 9x^2 - 5x + 7

Quadratic equations via completing the square

😀
Elementary
Solve the equation by completing the square:
u2+2u=3u^2 + 2u = 3

Real-life Quadratic Equations

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Elementary
After such a long journey, it's finally time to deal with real-life situations where quadratic equations arise. Solve each of these problems.
In the naughty garden problem, we arrived at the equation:
x(x+5)=36x(x+5) = 36

Problematic Perfect Square

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Intermediate
Complete the square:
7w2−11w7w^2 - 11w

Minimum and Maximum

Method
🤔
Intermediate
Find the minimum of the expression and the x value at which it is achieved:
4x2+28x+694x^2 + 28x + 69

Completing the square formula

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Advanced
Derive the general formula for completing the square for any quadratic trinomial:
Ax2+Bx+C,A≠0Ax^2 + Bx + C,\enspace A \neq 0

Not so straightforward

🤯
Advanced
When completing the square, we implicitly take roots. But taking a root can lead to different results:
4x2+2⋅2x⋅3+9(4x2)2+2⋅4x2⋅9+(9)2(±2x)2+2⋅2x⋅3+(±3)24x^2 + 2\cdot 2x \cdot 3 + 9 \\ \left(\sqrt{4x^2}\right)^2 + 2\cdot \sqrt{4x^2} \cdot \sqrt{9} + \left(\sqrt{9}\right)^2 \\ (\pm 2x)^2 + 2\cdot 2x \cdot 3 + (\pm 3)^2
Where do all these signs go, and why can we complete the square normally? Prove that we can ignore the ambiguity of taking roots and complete the square without messing with signs.
Completing the Square