GCSE MATHS • HIGHER TIER

Completing the Square Made Simple

Rewrite quadratic expressions, solve equations and identify turning points using completed-square form.

Start Learning
Practice Questions
Download Worksheet

Clear method

Exam practice

Worked proofs

Completing the Square in Four Moves

1

Half

Take half of the coefficient of x.

2

Square

Square the number you found.

3

Balance

Add and subtract that square so the expression stays equivalent.

4

Rewrite

Form a perfect square, simplify, then use it to solve or find the turning point

What You’ll Learn

Rewrite quadratic expressions in completed-square form.

Use the basic completing-the-square method.

Recognise common perfect-square patterns.

Solve quadratic equations by completing the square.

Find turning points from completed-square form.

Connect completing the square with quadratic graphs.

Key Rules

Half the x Coefficient

Take half of the coefficient of x

Square the Result

Square the halved value

Keep It Balanced

Add and subtract the same square

Form a Perfect Square

Rewrite the first three terms as a square

Remember ±

Use both square roots when solving

Find the Turning Point

From (x − h)² + k → turning point = (h, k)

Worked Proof: Completing the Square

Step-by-Step Method

  • 1. Take half of the x coefficient.
  • 2. Square it.
  • 3. Add and subtract that value.
  • 4. Rewrite as a perfect square.
  • 5. Simplify.

More Worked Examples

1

Solving by Completing the Square

x² + 6x + 5 = 0

(x + 3)² - 4 = 0
(x + 3)² = 4
x + 3 = ±2
Answer: x = -1 or x = -5

2

Finding Turning Points

Find the turning point of: y = x² + 8x + 12

Complete the square: y = (x + 4)² - 4
Turning point = (-4, -4)

3

Complete the square: x² + 4x + 1

Half of 4 = 2

x² + 4x + 4 - 4 + 1
(x + 2)² - 3
Answer: (x + 2)² - 3

4

Complete the square: x² + 8x + 7

Half of 8 = 4

x² + 8x + 16 - 16 + 7
Answer: (x + 4)² - 9
  • Common GCSE Mistakes
  • Forgetting to square half the coefficient.
  • Making sign errors.
  • Simplifying the constant terms incorrectly.
  • Forgetting ± when taking square roots.
  • Mixing up the turning-point coordinates.
  • Exam Tips
  • Always start by halving the coefficient of x.
  • Square the halved value before adding and subtracting it.
  • Keep every algebraic step visible to reduce sign errors.
  • When solving, remember both the positive and negative square roots.
  • In y = (x - h)² + k, read the turning point carefully as (h, k).
  • Check your completed-square form by expanding it back out.

Skills Used in Completing the Square

Quadratic Expressions

Algebraic Manipulation

Perfect Squares

Solving Equations

Square Roots

Quadratic Graphs

Ready to Practise?

Printable Worksheet

Download and print practice questions with answers.

Open Resource

Interactive Quiz

Test your knowledge with instant feedback and hints.

Open Resource

Exam-Style Questions

Timed questions to build confidence for the real exam.

Open Resource

Watch the Completing the Square Tutorial

Step-by-step walkthroughs of Completing the Square with clear methods and exam tips.

Watch Video

Frequently Asked Questions

It means rewriting a quadratic expression in the form (x + a)² + b.

Take half of the coefficient of x.

It keeps the expression equivalent while allowing the first three terms to form a perfect square.

Rewrite the quadratic as a square, isolate the squared expression and then take both positive and negative square roots.

The completed-square form reveals the coordinates of the turning point directly.

They make it easier to recognise and rewrite quadratic expressions accurately.