# Completing the square

## Introduction

Completing the square is a method of writing any quadratic expression in the form: $$(x + a)^2 + b$$ where $$a$$ and $$b$$ are real numbers.

For example, $$x^2 + 8x + 6$$ can be written as $$(x + 4)^2 - 10$$. You can show this is true by expanding and then simplifying the second expression: \begin{aligned} (x + 4)^2 - 10 & = (x + 4)(x + 4) - 10 \\ & = x^2 + 4x + 4x + 16 - 10 \\ & = x^2 + 8x + 6 \\ \end{aligned} We have arrived back at our original expression.

## How to complete the square

### Example 1

Write the expression $$x^2 + 6x + 2$$ in the form $$(x + a)^2 + b$$ where $$a$$ and $$b$$ are integers.

The first step in completing the square is to take the coefficient of the $$x$$ term and divide it by two. In this case we get $$6 ÷ 2 = 3$$. This gives us our value for $$a$$.
Now we know $$a = 3$$ the first part of our completed expression will look like $$(x + 3)^2$$. Expanding this gives us: \begin{aligned} (x + 3)^2 & = (x + 3)(x + 3) \\ & = x^2 + 3x + 3x + 9 \\ & = x^2 + 6x + 9 \\ \end{aligned} We now have our $$6x$$ term but instead of $$+ 2$$ as in our original expression we have $$+ 9$$. Our current expression is $$7$$ larger than it needs to be. This tells us we need to subtract $$7$$ to get back to where we want. So our value for $$b = -7$$.
So after completing the square we get: \begin{aligned} x^2 + 6x + 2 & = (x + 3)^2 - 7 \end{aligned} You can check your answer by expanding the right hand side as in the introduction and make sure the two sides balance.

### Example 2

Write the expression $$x^2 - 10x + 40$$ in the form $$(x + a)^2 + b$$ where $$a$$ and $$b$$ are integers.

Again we first halve the coefficient of the $$x$$ term to get our $$a$$ value of $$-5$$. Now we have: \begin{aligned} (x - 5)^2 & = (x - 5)(x - 5) \\ & = x^2 - 5x - 5x + 25 \\ & = x^2 - 10x + 25 \\ \end{aligned} This time our expression is $$15$$ smaller than it needs to be so we need add an extra $$15$$ to complete the square.
Now we know $$b = 15$$ we have: \begin{aligned} x^2 - 10x + 40 & = (x - 5)^2 + 15 \end{aligned}

## Worksheets to practise completing the square

Try this worksheet to practise your skills.