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Here is everything you need to know about completing the square for GCSE maths (Edexcel, AQA and OCR). You'll learn how to recognise a perfect square, complete the square on algebraic expressions, and tackle more difficult problems with the coefficient of x² ≠ 1.

You will also learn how to solve quadratic equations by completing the square, and how the completed square form links to graphs of quadratic equations.

Look out for completing the square worksheets and exam questions at the end.

What is completing the square?

A quadratic expression like x² + 4x + 4 is called a perfect square.

This is because it factorises to give (x + 2)( x+ 2), which can also be written as (x + 2)².

We can see this idea diagrammatically as follows:

Most quadratic expressions are not perfect squares, and cannot be written in this form as a single squared bracket. When we complete the square, we try to fit the expression to the closest possible perfect square, with a little bit added or subtracted to make things work.

Some expressions will have an 'extra' amount over from a perfect square, such as:

So we would write this expression in completed square form as:

Some expressions will be 'missing' an amount to make a perfect square, such as:

So we would write this expression in completed square form as:

While it's easy to see this using diagrams for quadratic expressions with small coefficients, we need a better method for expressions with larger coefficients. You may have noticed already that we divide the coefficient of x by 2 in order to work out the nearest perfect square.

Completing the square is a really useful method for solving quadratic equations; the quadratic formula for solving quadratic equations is based on it and can be derived by completing the square.

The completed square form of a quadratic expression is also really useful for identifying key points of quadratic functions, such as the maximum or minimum of a quadratic parabola (also called the vertex), without having to draw a graph. You can see this in the examples below.

You may sometimes see an expression in the form (x + a) referred to as a binomial.

How to complete the square

In order to complete the square:

1. Find the closest perfect square by dividing the coefficient of x by 2.

2. Expand the perfect square expression.

3. Compare the constant term in the perfect square to the original expression, and adjust as needed.

Completing the square worksheet

Get your free completing the square worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Completing the square worksheet

Get your free completing the square worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Related lessons on quadratic equations

Completing the square is part of our series of lessons to support revision on quadratic equations and solving equations. You may find it helpful to start with the main solving equations lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Completing the square examples

Example 1: perfect square

Complete the square for the expression

Find the closest perfect square by dividing the coefficient of x by 2.

The coefficient of x is 8, so when we divide this by 2, we get 4.

The closest perfect square is:

2 Expand the perfect square expression.

\[(x + 4)^{2} = x^{2} + 8x + 16\]

3 Compare the constant term in the perfect square to the original expression, and adjust as needed.

These match (because our example was a perfect square), so we don't need to make any adjustment.

The answer in complete square form is
Graphically
This graph shows the curve

The minimum value of y occurs when the bracket equals 0. This happens when x = −4.

If we substitute x = −4, we get:

\[y = (-4 + 4)^{2} = 0^{2} = 0\]

So the coordinates of the vertex, which is a minimum point, are (−4, 0).

Example 4: coefficient of x² = 1 and a negative coefficient of x

This is really straightforward - just remember that your perfect square bracket will need subtraction rather than addition in the middle.

Complete the square for the expression

The coefficient of x is −10, so when we divide this by 2, we get −5.

In order to make the constant term correct, we need to subtract 8, because 25 − 8 = 17.

Graphically

The minimum value of y occurs when the bracket equals 0. This happens when x = −5.

\[y = (5 - 5)^{2} - 8 = 0^{2} - 8 = -8\]

So the coordinates of the vertex, which is a minimum point, are (5, −8).

How to complete the square when coefficient of x² is not 1²

If the coefficient of x2 is not 1, we must first factorise to get an expression where the coefficient of x2 is 1. If the expression involves the term ax², then we take out a factor of a.

1. Factorise.

2. Complete the square on the expression inside the brackets: find the closest perfect square by dividing the coefficient of x by 2.

3. Expand the perfect square expression.

4. Compare the constant term in the perfect square to the original expression, and adjust as needed.

5. Multiply out the factorised value.

Completing the square when coefficient not 1 example

Example 6: coefficient of x² ≠ 1

Complete the square for the expression

Factorise.

We take out the common factor of 2, so the expression becomes

\[2\left(x^{2} + 4 x-5\right)\]

then complete the square on the expression inside the bracket.

The coefficient of x is 4, so when we divide this by 2, we get 2.

In order to make the constant term correct, we need to subtract 9, because 4 − 9 = −5.

\[2\left(x^{2}+4 x-5\right)\]

\[2\left((x+2)^{2}-9\right)\]

Graphically

The minimum value of y occurs when the bracket equals 0. This happens when x = −2.

\[y=2(x+2)^{2}-18=0^{2}-18=-18\]

So the coordinates of the vertex, which is a minimum point, are (−2, −18).

Note: we can also complete the square for expressions with a negative x² coefficient in the same way, by taking out a negative factor in the first step.

\[-\left(x^{2}-5 x+3\right)\]

and then completed as in Example 5.

How to solve quadratic equations using completing the square

Some quadratic equations can be solved by factorising, but most require either completing the square or the quadratic formula - in fact, the quadratic formula is derived from completing the square (see worksheet).

In order to solve quadratic equations using complete the square:

1. Make sure the equation is rearranged so that the right hand side equals 0 (if necessary).

2. Complete the square on the left hand side.

3. Rearrange and solve the resulting equation for x.

Solving using completing the square method examples

Common misconceptions

Incorrect sign in the middle of the perfect square bracket

The sign will always be the same sign as the coefficient of x.

Errors with fraction arithmetic

In particular, remember to square the numerator and denominator when squaring a fraction.

Going the wrong way when working out the adjustment

Remember to always work out how you get from the perfect square to the expression in the question.

Forgetting the ± sign when using the completing the square method to solve quadratics

Check carefully to make sure you haven't lost one of your solutions!

Practice completing the square questions

Completing the square GCSE questions

1. x^{2}+12x+19 can be written in the form

(x+a)^{2}+b where a and b are integers.

Work out the values of a and b .
(2 marks)

2. (a) x^{2}-6x+7 can be written in the form

(x+a)^{2}+b where a and b are integers.

Work out the values of a and b .

(b) Hence, or otherwise, solve the equation

x^{2}-6x+7=0

Give your answers in surd form.
(4 marks)

(a)

a = −3
(1)
b = −2
(1)
(b)

x = 3 + \sqrt{2}
(1)
x = 3 − \sqrt{2}
(1)

3. (a) 3x^{2}+12x-18 can be written in the form

a(x+b)^{2}+c where a and b are integers.

Work out the values of a, b and c .

(b) Hence, or otherwise, find the coordinates of the turning point of the graph of y=3x^{2}+12x-18
(4 marks)

(a)

a = 3
(1)
b = 2
(1)
c = −30
(1)
(b)

(−2, −30)
(1)

Learning checklist

You have now learned how to:

Complete the square for a quadratic expression

Solve quadratic equations by completing the square

Identify turning points by completing the square

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Free completing the square GCSE maths revision guide, including step by step examples, exam questions and free worksheet.

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