Completing the Square: GCSE Maths Step-by-Step Guide

Completing the square is a core GCSE Higher Maths technique that lets you rewrite any quadratic in the form (x + p)² + q. Once you can do that, solving equations, finding turning points on quadratic graphs, and spotting minimum or maximum values all become straightforward. This guide walks through every step - with worked examples, practice questions, and exam tips - so you're ready for whatever AQA, Edexcel and OCR put in front of you.

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Table of Contents

1. What Is Completing the Square?
2. Why Use Completed Square Form?
3. The Method - Step by Step
4. Worked Example 1 - x² + 6x + 5
5. Worked Example 2 - x² − 8x + 3
6. When the Coefficient of x² Is Not 1
7. Worked Example 3 - 2x² + 12x − 4
8. Completing the Square Examples - Quick Reference
9. Solving Quadratic Equations by Completing the Square
10. Finding the Turning Point of a Quadratic Graph
11. Minimum and Maximum Points
12. Practice Questions
13. Common Mistakes
14. Exam Tips
15. Frequently Asked Questions
16. Completing the Square vs Other Methods

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Introduction

If you're a GCSE student working through GCSE higher algebra and you've hit the topic of completing the square GCSE for the first time, you're in the right place. This is one of those techniques that looks intimidating at first glance - but once the logic clicks, it's genuinely one of the most satisfying things in higher maths. For a Higher level student, mastering this single method unlocks three separate skills at once.

Completing the square examples appear throughout GCSE Maths papers, so it's worth getting comfortable with the full range of question types. The method lets you rewrite a quadratic expression in a neater form - and that neater form is the key to solving quadratic equations GCSE questions that won't factorise, finding the turning point of a parabola, and identifying whether that point is a minimum or a maximum.

All three skills come up regularly in GCSE maths exams, and all three follow directly from the same method.

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What Is Completing the Square?

Completing the square is an important algebra technique in GCSE Higher Maths. It is one of the core methods for working with quadratic equations - alongside factorising and the quadratic formula.

It allows you to rewrite a quadratic expression in a form that makes several things easier to see.

You can use completing the square to:

• Rewrite quadratic expressions
• Solve quadratic equations
• Find the turning point of a parabola
• Identify minimum or maximum values
• Work with quadratics that do not factorise easily

A quadratic such as:

x² + 6x + 5

can be rewritten as:

(x + 3)² − 4

Both expressions have exactly the same value. They are simply written in different forms.

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Why Use Completed Square Form?

The completed square form

(x + p)² + q

is useful because:

• It reveals the turning point of the quadratic graph directly - the coordinates are (−p, q).
• It lets you solve quadratic equations that cannot be factorised by taking the square root of both sides.
• It connects directly to the quadratic formula, which is itself derived by completing the square on ax² + bx + c = 0.
• It may be required explicitly in GCSE Higher questions where the completed square form is part of the working.

For GCSE students aiming for higher grades, this is an important method to understand.

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The Method - Step by Step

This method works when the coefficient of x² is 1.

Starting expression:

x² + bx + c

Step 1

Write a squared bracket:

(x + ___)²

Step 2

Inside the bracket, write half the coefficient of x.

Step 3

Remember that:

(x + b/2)² = x² + bx + (b/2)²

Step 4

The bracket introduces an extra:

(b/2)²

Subtract this outside the bracket.

Step 5

Simplify the constant terms.

General Result

(x + b/2)² − (b/2)² + c

This is the completed square form.

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Worked Example 1 - x² + 6x + 5

Write x² + 6x + 5 in completed square form.

Final Answer

(x + 3)² − 4

Check

(x + 3)² − 4

= x² + 6x + 9 − 4

= x² + 6x + 5 ✓

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Worked Example 2 - x² − 8x + 3

Write x² − 8x + 3 in completed square form.

Half of −8 is:

−4

Write:

(x − 4)²

Expand:

(x − 4)² = x² − 8x + 16

The bracket introduces an extra +16, so subtract 16.

Then add the original constant:

(x − 4)² − 16 + 3

Simplify:

(x − 4)² − 13

Final Answer

(x − 4)² − 13

Check

(x − 4)² − 13

= x² − 8x + 16 − 13

= x² − 8x + 3 ✓

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When the Coefficient of x² Is Not 1

When the quadratic expression is:

ax² + bx + c

with a ≠ 1, you need one extra step.

Step 0

Factor out a from the x² and x terms.

Leave the constant outside.

Then complete the square inside the brackets and simplify.

This is a common source of errors, so take care when factoring.

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Worked Example 3 - 2x² + 12x − 4

Write 2x² + 12x − 4 in completed square form.

Step 1

Factor out 2 from the first two terms:

2(x² + 6x) − 4

Step 2

Complete the square inside the bracket.

Half of 6 is 3:

(x + 3)² − 9

Therefore:

2[(x + 3)² − 9] − 4

Step 3

Expand the outer factor:

2(x + 3)² − 18 − 4

Step 4

Simplify:

2(x + 3)² − 22

Final Answer

2(x + 3)² − 22

Check

2(x + 3)² − 22

= 2(x² + 6x + 9) − 22

= 2x² + 12x + 18 − 22

= 2x² + 12x − 4 ✓

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Completing the Square Examples - Quick Reference

Use this table as a quick revision guide.

The turning point coordinates can be read directly from the completed square form.

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Solving Quadratic Equations by Completing the Square

One of the most important uses of this method is to solve quadratic equations, particularly those that do not factorise neatly.

To solve a quadratic equation by completing the square:

1. Rearrange the equation so it is equal to 0.
2. Complete the square.
3. Isolate the squared bracket.
4. Take the square root of both sides.
5. Remember the ±.
6. Solve for x.

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Worked Example - Solve x² + 6x + 5 = 0

We already know:

x² + 6x + 5 = (x + 3)² − 4

Therefore:

(x + 3)² − 4 = 0

Add 4 to both sides:

(x + 3)² = 4

Take the square root:

x + 3 = ±2

Therefore:

x = −3 + 2 = −1

or

x = −3 − 2 = −5

Solutions

x = −1 or x = −5

Check

For x = −1:

(−1)² + 6(−1) + 5 = 0 ✓

For x = −5:

(−5)² + 6(−5) + 5 = 0 ✓

Exam Tip: Check your solutions by substituting them back into the original equation.

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Worked Example - Solve x² − 8x + 3 = 0

We already know:

x² − 8x + 3 = (x − 4)² − 13

Therefore:

(x − 4)² − 13 = 0

Add 13:

(x − 4)² = 13

Take the square root:

x − 4 = ±√13

Therefore:

x = 4 + √13

or

x = 4 − √13

Exact Solutions

x = 4 + √13 or x = 4 − √13

If decimal answers are required:

x ≈ 7.61 or x ≈ 0.39

Note: When a quadratic expression doesn't factorise, completing the square can be a very useful method. The quadratic formula is another option.

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Finding the Turning Point of a Quadratic Graph

The turning point of a quadratic graph, or parabola, is the point where the curve changes direction.

It may change:

• From decreasing to increasing - a minimum
• From increasing to decreasing - a maximum

Completed square form makes the turning point easy to identify.

If:

y = (x + p)² + q

then the turning point is:

(−p, q)

No substitution or differentiation is needed.

Why Does This Work?

The expression:

(x + p)²

is always greater than or equal to 0.

Its smallest possible value is:

0

This occurs when:

x = −p

At that point:

y = q

Therefore, the turning point is:

(−p, q)

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Worked Example - Turning Point of y = x² + 6x + 5

We know:

y = (x + 3)² − 4

Therefore, the turning point is:

(−3, −4)

The coefficient of x² is positive, so the parabola opens upward.

Therefore, this is a:

Minimum point

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Minimum and Maximum Points

The sign of the coefficient of x² tells you whether the turning point is a minimum or maximum.

Worked Example - Maximum Point

Find the maximum point of:

y = −(x − 2)² + 7

Here:

a = −1

The coefficient is negative, so the parabola opens downward.

The turning point is:

(2, 7)

Therefore, it is a:

Maximum point

The maximum value of the function is:

y = 7

when:

x = 2

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Practice Questions

Try these yourself before checking the answers.

Question 1

Write:

x² + 10x + 3

in completed square form.

Question 2

Write:

x² − 4x − 7

in completed square form.

Question 3

Write:

3x² + 6x − 9

in completed square form.

Question 4

Solve:

x² + 10x + 3 = 0

Give your answers in exact form.

Question 5

Find the turning point of:

y = x² − 4x − 7

State whether it is a minimum or maximum point.

Question 6

A quadratic expression is written as:

(x − 5)² − 11

Write it in the form:

x² + bx + c

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Answers

Question 1

Half of 10 is 5:

(x + 5)² − 25 + 3

Therefore:

(x + 5)² − 22

Question 2

Half of −4 is −2:

(x − 2)² − 4 − 7

Therefore:

(x − 2)² − 11

Question 3

Factor out 3:

3(x² + 2x) − 9

Half of 2 is 1:

3[(x + 1)² − 1] − 9

Simplify:

3(x + 1)² − 3 − 9

Therefore:

3(x + 1)² − 12

Question 4

(x + 5)² − 22 = 0

Therefore:

(x + 5)² = 22

Take the square root:

x + 5 = ±√22

Therefore:

x = −5 + √22

or

x = −5 − √22

Question 5

Complete the square:

y = (x − 2)² − 11

Turning point:

(2, −11)

The coefficient of x² is positive.

Therefore, this is a:

Minimum point

Question 6

Expand:

(x − 5)² − 11

= x² − 10x + 25 − 11

= x² − 10x + 14

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Common Mistakes

These are some of the errors that can cost GCSE students marks.

1. Forgetting to Subtract the Square

After writing:

(x + p)²

you must subtract:

p²

outside the bracket.

Skipping this step changes the value of the expression.

2. Sign Errors With Negative x Coefficients

For:

x² − 8x

half of −8 is −4.

Therefore, write:

(x − 4)²

not:

(x + 4)²

3. Forgetting ± When Taking the Square Root

If:

(x + p)² = k

then:

x + p = ±√k

Forgetting the negative root means you may only find one solution.

4. Wrong Sign on the Turning Point

If the completed square form is:

(x + 3)² − 4

the turning point is:

(−3, −4)

not:

(3, −4)

The x-coordinate has the opposite sign to the number inside the bracket.

5. Incorrect Factoring When a ≠ 1

For:

2x² + 12x − 4

factor 2 from the x² and x terms:

2(x² + 6x) − 4

Do not incorrectly change the value of the constant.

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Exam Tips

These tips apply across AQA Maths, Edexcel Maths, and OCR Maths.

1. Show every step. Correct working can earn method marks even if an arithmetic error occurs later.

2. Check your completed square form. Expand it back and confirm it matches the original expression.

3. Remember ± when square rooting. Forgetting the negative solution is a common mistake.

4. State the turning point explicitly. Write the coordinates clearly, such as (−3, −4).

5. Watch the sign inside the bracket. If the form is (x + 3)², the turning point x-coordinate is −3.

6. When a ≠ 1, factor carefully. Factor out the coefficient of x² before completing the square.

7. Use exact values unless told otherwise. If the question asks for an exact solution, leave answers such as √13 rather than converting them to decimals.

8. Practise using GCSE Maths past papers. This helps you become familiar with the wording and structure of completing-the-square questions.

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Frequently Asked Questions

What Is Completing the Square in GCSE Maths?

Completing the square is an algebraic method for rewriting a quadratic expression:

ax² + bx + c

in the form:

a(x + p)² + q

This form makes it easier to:

• Solve quadratic equations
• Find the turning point
• Determine whether the turning point is a minimum or maximum

When Do I Use Completing the Square Instead of Factorising?

Use completing the square when:

• The quadratic does not factorise neatly
• The question asks for completed square form
• The question asks for the turning point of a quadratic graph

How Do I Find the Turning Point Using Completing the Square?

Write the quadratic in the form:

(x + p)² + q

The turning point is:

(−p, q)

If the coefficient of x² is positive, it is a minimum.

If the coefficient of x² is negative, it is a maximum.

Does Completing the Square Work for All Quadratic Equations?

Yes.

Every quadratic expression can be written in completed square form.

The method works whether or not the quadratic factorises and whether or not the solutions are rational.

What Is the Difference Between Completing the Square and the Quadratic Formula?

Both methods can be used to solve quadratic equations.

The quadratic formula is derived by completing the square on:

ax² + bx + c = 0

Completing the square also has the advantage of allowing you to identify the turning point directly.

Is Completing the Square on AQA, Edexcel and OCR GCSE Specifications?

Completing the square is a Higher-tier GCSE Maths topic across the major exam boards, including:

• AQA
• Edexcel
• OCR

What Does "Equal to 0" Mean When Solving by Completing the Square?

When solving a quadratic equation, you may first rearrange it so that:

ax² + bx + c = 0

Then complete the square on the left-hand side, isolate the squared bracket, and take the square root of both sides.

How Do I Handle a Negative Coefficient of x²?

Factor out the negative coefficient first.

For example:

−x² + 4x − 1

can be written as:

−(x² − 4x + 1)

Then complete the square inside the bracket.

A negative coefficient of x² means the parabola opens downward, so its turning point is a maximum.

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Completing the Square vs Other Methods

There are three main approaches to solving quadratic equations at GCSE level.

If a question about a quadratic graph asks for the minimum or maximum point, completing the square is often the most useful method.

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Key Points to Remember

• Completing the square rewrites a quadratic into a(x + p)² + q form.
• Take half the coefficient of x and place it inside the squared bracket.
• Subtract the square of that value outside the bracket.
• If the coefficient of x² is not 1, factor it out first.
• The turning point of y = (x + p)² + q is (−p, q).
• A positive coefficient of x² gives a minimum point.
• A negative coefficient of x² gives a maximum point.
• Remember ± when taking square roots.
• Check your completed square form by expanding it.