Skill 5 of 12

Solve quadratic equations by completing the square

Solve quadratic equations with real solutions by completing the square and taking square roots.

Common Core standards: A-REI.4aA-REI.4b
Builds on: Complete the square to write vertex form
Solve quadratic equations using square roots

Common mistakes

What students most often get wrong with this skill:

Adding $(b/2)^2$ to only one side of the equation
Forgetting plus-or-minus after taking the square root
Arithmetic errors when moving the constant term

A worked example

From the lesson, with every step explained.

Example 1: rational solutions

Problem. Solve $x^2+4x-5=0$ .

Plan. Move the constant, complete the square, then take square roots.

1) Add 5 to both sides to move the constant.

x^2+4x=5

2) Here $b=4$ . Half of $b$ is $2$ , and $2^2=4$ . Add 4 to both sides.

x^2+4x+4=5+4

3) The left side is now a perfect square, $(x+2)^2$ .

(x+2)^2=9

4) Take square roots of both sides. We need $\pm$ because both $3$ and $-3$ square to 9.

x+2=\pm 3

5) Subtract 2 and split into two cases.

x=-2+3=1

x=-2-3=-5

Answer. $x=1$ or $x=-5$ .

Check. Substitute each value.

1^2+4(1)-5=0

(-5)^2+4(-5)-5=25-20-5=0

Both work.

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