Expanding & factorising quadratics – GCSE Maths

Foundation and Higher · Grade 5 · Non-calculator

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Multiply out two brackets to get a quadratic, and reverse it by factorising $x^{2} + b x + c$ and the difference of two squares. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page.

Double brackets expand to a quadratic: $(2 x + 3) (x - 5) = 2 x^{2} - 7 x - 15$ . A grid of four products stops you missing the middle terms. Factorising $a x^{2} + b x + c$ when $a = 1$ is two numbers that multiply to $c$ and add to $b$ . When $a \neq = 1$ , find two numbers that multiply to $a c$ and add to $b$ , then split the middle term and factor by grouping — or use inspection with the first terms multiplying to $a x^{2}$ .

Difference of two squares: $x^{2} - 16 = (x - 4) (x + 4)$ , and $4 x^{2} - 9 = (2 x - 3) (2 x + 3)$ . Expand in the margin every time. An $x$ common factor comes out first: $2 x^{2} + 6 x = 2 x (x + 3)$ .

Expand with four products (or a grid). Factorise $x^{2} + b x + c$ by two numbers that multiply to $c$ and add to $b$ . Difference of two squares: $a^{2} - b^{2} = (a - b) (a + b)$ . Check by expanding back.

Objectives

Expand two brackets.
Expand a squared bracket.
Factorise a quadratic and a difference of squares.

Key points

Expand $(x + a) (x + b)$ using four multiplications and collect the two $x$ terms.
Expand and simplify expressions such as $(x + 4)^{2}$ and $(x + 4)^{2} - (x - 2) (x + 2)$ .
Factorise $x^{2} + b x + c$ by finding two numbers that multiply to $c$ and add to $b$ .
Recognise and factorise the difference of two squares $a^{2} - b^{2} = (a + b) (a - b)$ .

Worked examples

Example 1

Expand (x+3)(x+2).

2 marks

Multiply every term: x(x) + x(2) + 3(x) + 3(2).
Collect like terms: x^2 + 2x + 3x + 6 = x^2 + 5x + 6.
The expanded quadratic is x^2+5x+6.

Answer: x^2+5x+6

Common slips

Slip: Multiply only the first and last terms. FOIL requires four multiplications for two binomials.
Slip: Vera wrote x^2 + 6. The cross terms 2x + 3x sum to 5x.
Slip: Conclude x^2+6. Collecting all terms produces x^2+5x+6.

Example 2

Expand (x+4)^2.

2 marks

Write as repeated brackets: (x+4)(x+4).
Expand and collect: x^2 + 4x + 4x + 16 = x^2 + 8x + 16.
The expanded quadratic is x^2+8x+16.

Answer: x^2+8x+16

Common slips

Slip: Square each term individually to get x^2 + 16. (x+4)^2 produces a cross term 2(4x) = 8x.
Slip: Will wrote x^2 + 16. The middle term is twice the product of x and 4.
Slip: Conclude x^2+16. Doubling the product gives 8x in the middle.

Example 3

Factorise x^2-9.

3 marks

Recognise difference of two squares: a^2 - b^2 = (a-b)(a+b).
Write factors using square root 3: (x-3)(x+3).
The factorised form is (x-3)(x+3).

Answer: (x-3)(x+3)

Common slips

Slip: Factorise as (x-9)(x+1). 9 is 3 squared, so the square root is 3.
Slip: Xena wrote (x-9)(x+1). Difference of squares uses the square root of 9, which is 3.
Slip: Conclude (x-9)(x+1). (x-3)(x+3) expands back to x^2 - 9.

Questions and answers

GCSE Maths expanding & factorising quadratics questions and answers. Try each one, then open the answer. Easy questions come first.

Question 1 Easy2 marks

Expand (x+4)(x+1).

Show answer

Answer: x^2+5x+4

Multiply out: x^2 + x + 4x + 4.
Collect terms: x^2 + 5x + 4.

M1 — Multiply out: x^2 + x + 4x + 4.
A1 — Collect terms: x^2 + 5x + 4.

Question 2 Easy2 marks

Expand (x+5)(x+2).

Show answer

Answer: x^2+7x+10

Multiply out: x^2 + 2x + 5x + 10.
Collect terms: x^2 + 7x + 10.

M1 — Multiply out: x^2 + 2x + 5x + 10.
A1 — Collect terms: x^2 + 7x + 10.

Question 3 Medium2 marks

Expand (x+3)^2.

Show answer

Answer: x^2+6x+9

Repeated brackets: (x+3)(x+3).
Middle 2×3=6, constant 9: x^2+6x+9.

M1 — Repeated brackets: (x+3)(x+3).
A1 — Middle 2×3=6, constant 9: x^2+6x+9.

Question 4 Medium2 marks

Expand (x+5)^2.

Show answer

Answer: x^2+10x+25

Repeated brackets: (x+5)(x+5).
Middle 2×5=10, constant 25: x^2+10x+25.

M1 — Repeated brackets: (x+5)(x+5).
A1 — Middle 2×5=10, constant 25: x^2+10x+25.

Question 5 Hard2 marks

Factorise x^2-16.

Show answer

Answer: (x-4)(x+4)

Difference of squares with √16 = 4.
Write factors: (x-4)(x+4).

M1 — Difference of squares with √16 = 4.
A1 — Write factors: (x-4)(x+4).

Question 6 Hard2 marks

Factorise x^2-25.

Show answer

Answer: (x-5)(x+5)

Difference of squares with √25 = 5.
Write factors: (x-5)(x+5).

M1 — Difference of squares with √25 = 5.
A1 — Write factors: (x-5)(x+5).

Question 7 Exam-style2 marks

Expand (x+6)^2.

Show answer

Answer: x^2+12x+36

Write as repeated brackets: (x+6)(x+6).
Compute middle term 2(6)x = 12x and constant 36: x^2 + 12x + 36.

M1 — x^2 + 6x + 6x + 36, with at least three terms correct
A1 — x^2 + 12x + 36

Common mistakes

Most lost marks on this topic are one slip.

Dropping the middle term of a pair or a square
Add the two numbers for the x term, or double the number in a square
Using the same sign in a difference of squares
One factor adds the root and one subtracts it
Leaving 2a+2b without the common factor
Take out 2

Spot the slip

Ada expands (x+3)(x+2) and writes x^2+6. What is the full expansion?

x^2+5x+6
x^2+6
x^2+6x+5

Show the slip

x^2+5x+6 The x term is 3+2 = 5x, and constant is 3×2 = 6, giving x^2+5x+6.

Exam tips

Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write all four products on one line before collecting — a marker can give the method mark even if a sign slips later.
For factorising, jot the factor pairs of $c$ in the margin; it is quicker than guessing and shows your method.

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

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Questions students ask

Is Expanding & factorising quadratics Foundation or Higher?

Expanding & factorising quadratics is on Foundation and Higher, aimed at grade 5.

What is a common slip on Expanding & factorising quadratics?

Dropping the middle term of a pair or a square. Add the two numbers for the x term, or double the number in a square.

Where can I get Expanding & factorising quadratics questions and answers?

The questions on this page are free. Open Show answer under each one, or download the worksheet PDF. Start free to practise this topic in the app.

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AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.

Expanding & factorising quadratics – GCSE Maths

Objectives

Key points

Worked examples

Example 1

Example 2

Example 3

Questions and answers

Question 1 Easy2 marks

Question 2 Easy2 marks

Question 3 Medium2 marks

Question 4 Medium2 marks

Question 5 Hard2 marks

Question 6 Hard2 marks

Question 7 Exam-style2 marks

Common mistakes

Spot the slip

Exam tips

Printable worksheet

Questions students ask

Is Expanding & factorising quadratics Foundation or Higher?

What is a common slip on Expanding & factorising quadratics?

Where can I get Expanding & factorising quadratics questions and answers?

Related topics

More on Expanding & factorising quadratics

Practise Expanding & factorising quadratics free