Completing the Square Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Solve x2−4x−5=0x^2 - 4x - 5 = 0 by completing the square.

Solution

  1. 1
    Move the constant: x2−4x=5x^2 - 4x = 5.
  2. 2
    Half of −4-4 is −2-2; square it: 44. Add to both sides: x2−4x+4=9x^2 - 4x + 4 = 9.
  3. 3
    Factor: (x−2)2=9(x - 2)^2 = 9.
  4. 4
    Take square roots: x−2=±3x - 2 = \pm 3, giving x=5x = 5 or x=−1x = -1.

Answer

x=5 or x=−1x = 5 \text{ or } x = -1
Completing the square can also solve equations. After creating a perfect square on the left, take square roots of both sides.

About Completing the Square

A technique for rewriting ax2+bx+cax^2 + bx + c in vertex form a(x−h)2+ka(x - h)^2 + k by adding and subtracting the value (b2a)2\left(\frac{b}{2a}\right)^2 to create a perfect square trinomial.

Learn more about Completing the Square →

More Completing the Square Examples

Example 1 medium

Rewrite [formula] in vertex form by completing the square.

Example 3 easy

What number completes the square for [formula]?

Example 4 medium

Rewrite [formula] in vertex form.

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