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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Visualizing the process of completing the square
Special case: x2+bx=(x+b2)2−(b2)2x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2

Completing the square

The process of “packing” an expanded expression into a squared binomial (a parenthesis squared):
Examples:
x2+2x+1⇒(x+1)2x^2 + 2x + 1 \Rightarrow \green{(x + 1)^2}
16−8y+y2⇒(4−y)2 16 - 8y + y^2 \Rightarrow \green{(4 - y)^2}
t2−t⇒(t−12)2−14 t^2 - t \Rightarrow \green{\left(t - \frac{1}{2}\right)^2} - \frac{1}{4}
Why complete the square?
  • 1
    It allows solving any quadratic equation.
  • 2
    To simplify expressions. Before completing the square, the unknown appears twice in the expression — to the second and first power.
    9x2One+6xTwo+19\underset{\text{One}}{\red{x^2}} + 6\underset{\text{Two}}{\red{x}} + 1
    After completing the square, the unknown appears only once. This can be useful not only for solving equations but also for other tasks.
    (3xOne+1)2(3\underset{\text{One}}{\brand{x}} + 1)^2
    • Sometimes you can complete the square in one go. This works when the expression matches the a2±2ab+b2a^2 \pm 2ab + b^2 pattern from special products:

    Completing the square

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    Example
    Algebraically complete the square in the expression:
    x2+10x+25x^2 + 10x + 25
    But often the expression is incomplete. Then you have to add the missing parts yourself, but don't forget to compensate:

    Completing the square with incomplete terms

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    Example
    Complete the square in the expression by adding and compensating terms:
    x2−xx^2-x
    Completing the square allows solving any quadratic equation:

    Quadratic equations via completing the square

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    Example
    Solve the quadratic equation by completing the square:
    x2−2x−35=0x^2 - 2x - 35 = 0
    Completing the Square