Solve x² + 4x - 12 = 0 by completing the square.

(x + 2)² = 16, so x = 2 or x = -6

Solve x² - 6x + 4 = 0, giving exact answers.

(x - 3)² = 5, so x = 3 ± √5

Free worksheet · Australian Curriculum

Completing the square worksheets with answers

Completing the square turns any quadratic into a perfect square plus a number, which lets you solve equations that won't factorise and find the turning point of a parabola.

Make a worksheet ↓How to do it and a quick check ↓

Topic guide

Completing the square, explained

The one step at the heart of it, then how to use it to solve equations and sketch parabolas.

1. Perfect squares

A perfect square expands like this:

$(x + a)^2 = x^2 + 2ax + a^2$

The number in front of $x$ is $2a$ , and the constant is $a^2$ . So the constant is always the square of half the $x$ coefficient.

2. The key step: halve, then square

To make $x^2 + 6x$ into a perfect square, halve $6$ to get $3$ , then square it to get $9$ :

$x^2 + 6x + 9 = (x + 3)^2$

3. Completing the square on any quadratic

Add and subtract the number you need, so the expression doesn't change:

$x^2 + 6x + 5 = x^2 + 6x + 9 - 9 + 5 = (x + 3)^2 - 4$

4. Solving an equation

Solve $x^2 + 6x + 5 = 0$ . Complete the square, move the number across, then take the square root of both sides (positive and negative):

$(x + 3)^2 = 4, x + 3 = \pm 2, x = -1 \text{ or } x = -5$

It also works when the answers are surds. For $x^2 + 4x - 1 = 0$ , $(x + 2)^2 = 5$ , so $x = -2 \pm \sqrt{5}$ .

5. Turning point form

Completing the square writes a parabola as $y = (x - h)^2 + k$ , with its turning point at $(h, k)$ .

$y = x^2 - 8x + 10 = (x - 4)^2 - 6$

So the turning point is $(4, -6)$ . Watch the sign: $(x - 4)$ means $h = 4$ .

6. When the x² coefficient isn't 1 (Year 10A)

Take the coefficient out of the $x$ terms first, then complete the square inside the brackets:

$2x^2 + 12x + 7 = 2(x^2 + 6x) + 7 = 2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11$

Quick check: 10 questions

Try each one, then open the answer.

What number completes the square: $x^2 + 10x + \underline{ }$ ?
Answer
$25$
What number completes the square: $x^2 - 6x + \underline{ }$ ?
Answer
$9$
Write $x^2 + 8x + 3$ in the form $(x + a)^2 + b$ .
Answer
$(x + 4)^2 - 13$
Write $x^2 - 2x - 7$ in the form $(x + a)^2 + b$ .
Answer
$(x - 1)^2 - 8$
What number completes the square: $x^2 + 5x + \underline{ }$ ?
Answer
$\dfrac{25}{4}$
Solve $x^2 + 4x - 12 = 0$ by completing the square.
Answer
$(x + 2)^2 = 16$ , so $x = 2$ or $x = -6$
Solve $x^2 - 6x + 4 = 0$ , giving exact answers.
Answer
$(x - 3)^2 = 5$ , so $x = 3 \pm \sqrt{5}$
Find the turning point of $y = x^2 + 2x - 3$ .
Answer
$y = (x + 1)^2 - 4$ , so $(-1, -4)$
Write $x^2 + 3x + 1$ in the form $(x + a)^2 + b$ .
Answer
$\left(x + \dfrac{3}{2}\right)^2 - \dfrac{5}{4}$
Write $2x^2 - 8x + 3$ in the form $a(x - h)^2 + k$ .
Answer
$2(x - 2)^2 - 5$

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These topic pages cover this skill, with worked examples, common mistakes and a question to try.

Year 10 worksheetQuadratic equations Year 10 worksheetQuadratic factorising Year 10A worksheetFactorising non-monic quadratics Year 10 worksheetGraphs of parabolas, circles, hyperbolas and exponentials

Questions

What does completing the square mean?

It means rewriting a quadratic as a perfect square plus or minus a number, for example x squared plus 6x plus 5 becomes (x plus 3) squared minus 4.

Why use completing the square instead of factorising?

It works on every quadratic, including ones that don't factorise neatly, and it shows the turning point of the parabola straight away.

What year is completing the square taught?

It is usually taught in Year 10 alongside solving quadratic equations, and used again in senior maths. Quadratics with a leading coefficient other than 1 are Year 10A.

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