Completing the Square Practice Worksheet

Practice rewriting quadratics and solving equations by completing the square with step-by-step solutions.

Try each problem first. Then click Show solution to check your work. Completing the square is all about creating a perfect square trinomial.

Key Idea

For an expression like \(x^2 + bx\), add:

\[ \left(\frac{b}{2}\right)^2 \]
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Level 1: Build the Perfect Square

Problem 1

Complete the square:

\[ x^2 + 6x \]
Show solution

Take half of \(6\), then square it.

\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \]

Add \(9\) to make a perfect square trinomial.

\[ x^2 + 6x + 9 = (x+3)^2 \]
Problem 2

Complete the square:

\[ x^2 - 10x \]
Show solution

Take half of \(-10\), then square it.

\[ \left(\frac{-10}{2}\right)^2 = (-5)^2 = 25 \]

Add \(25\) to make a perfect square trinomial.

\[ x^2 - 10x + 25 = (x-5)^2 \]
Problem 3

Complete the square:

\[ x^2 + 14x \]
Show solution

Take half of \(14\), then square it.

\[ \left(\frac{14}{2}\right)^2 = 7^2 = 49 \]
\[ x^2 + 14x + 49 = (x+7)^2 \]

Level 2: Solve by Completing the Square

Problem 4

Solve:

\[ x^2 + 6x = 16 \]
Show solution

Complete the square by adding \(9\) to both sides.

\[ x^2 + 6x + 9 = 16 + 9 \] \[ (x+3)^2 = 25 \]

Take the square root of both sides.

\[ x+3 = \pm 5 \] \[ x = -3 \pm 5 \] \[ x = 2,\ -8 \]
Problem 5

Solve:

\[ x^2 - 8x = 9 \]
Show solution

Half of \(-8\) is \(-4\), and \((-4)^2=16\).

\[ x^2 - 8x + 16 = 9 + 16 \] \[ (x-4)^2 = 25 \]

Take the square root.

\[ x-4 = \pm 5 \] \[ x = 4 \pm 5 \] \[ x = 9,\ -1 \]
Problem 6

Solve:

\[ x^2 + 4x = 12 \]
Show solution

Half of \(4\) is \(2\), and \(2^2=4\).

\[ x^2 + 4x + 4 = 12 + 4 \] \[ (x+2)^2 = 16 \]
\[ x+2 = \pm 4 \] \[ x = -2 \pm 4 \] \[ x = 2,\ -6 \]

Level 3: Standard Form Quadratics

Problem 7

Solve:

\[ x^2 + 10x + 21 = 0 \]
Show solution

Move the constant to the other side.

\[ x^2 + 10x = -21 \]

Half of \(10\) is \(5\), and \(5^2=25\).

\[ x^2 + 10x + 25 = -21 + 25 \] \[ (x+5)^2 = 4 \]
\[ x+5 = \pm 2 \] \[ x = -5 \pm 2 \] \[ x = -3,\ -7 \]
Problem 8

Solve:

\[ x^2 - 12x + 20 = 0 \]
Show solution

Move the constant to the other side.

\[ x^2 - 12x = -20 \]

Half of \(-12\) is \(-6\), and \((-6)^2=36\).

\[ x^2 - 12x + 36 = -20 + 36 \] \[ (x-6)^2 = 16 \]
\[ x-6 = \pm 4 \] \[ x = 6 \pm 4 \] \[ x = 10,\ 2 \]

Need more help with completing the square?

Completing the square is easier when you understand why \(\left(\frac{b}{2}\right)^2\) creates a perfect square. For a full explanation, visit my completing the square guide.

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