A4AlgebraFoundation & HigherGrades 2–7

Expanding Brackets

Q: How do I know I have expanded correctly?

Substitute a simple value such as x = 2 into the original and your answer. If they give different numbers, there is a mistake.

Expanding means multiplying out brackets so the expression is written without them. Every term inside a bracket is multiplied by whatever is outside — no term gets left behind. It is the reverse of factorising, and it is the first step in solving most equations with brackets.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

Expanding is really about area. A rectangle $3$ wide and $(x + 4)$ long has area $3 (x + 4)$ , but you can also split it into two pieces, $3 x$ and $12$ . Both describe the same area, so $3 (x + 4) \equiv 3 x + 12$ .

Double brackets work the same way with a rectangle split into four pieces. Seeing it as a grid is the most reliable method — it stops terms going missing, which is where most marks are lost.

Expanding Single BracketsCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Expand a single bracket, including negative terms outside
Expand and simplify expressions with more than one bracket
Expand the product of two binomials, including squared brackets
Expand the product of three binomials (Higher)

Single brackets

Multiply every term inside the bracket by the term outside: $a (b + c) = ab + a c$ .

Take care with signs. A negative outside changes the sign of every term inside: $- 2 (x - 5) = - 2 x + 10$ .

When there are two brackets, expand each one, then collect like terms.

4 \times 3 x = 12 x

4 \times (- 5) = - 20

Answer

12 x - 20

Your turnSame method, new numbers — try it before moving on.

Expand $6 (2 y + 7)$ .

Your answer

Your turnSame method, new numbers — try it before moving on.

Expand and simplify $5 (x + 3) - 3 (x - 2)$ .

Your answer

Tip — Draw arrows from the outside term to each inside term before you write anything — one arrow per term.

Double brackets

Every term in the first bracket multiplies every term in the second. Two terms times two terms gives four products, which usually collapse to three terms after collecting.

A grid keeps the four products organised. FOIL — First, Outer, Inner, Last — is the same thing written in a line.

(x + 3) (x + 2) = x^{2} + 3 x + 2 x + 6 = x^{2} + 5 x + 6

1First:

x \times x = x^{2}

. Outer:

x \times (- 3) = - 3 x

2Inner:

5 \times x = 5 x

. Last:

5 \times (- 3) = - 15

3Collect the

x

terms:

- 3 x + 5 x = 2 x

Answer

x^{2} + 2 x - 15

Your turnSame method, new numbers — try it before moving on.

Expand and simplify $(x - 4) (x - 6)$ .

Your answer

Type x² as x^2.

Your turnSame method, new numbers — try it before moving on.

Expand and simplify $(3 x - 2) (2 x + 5)$ .

Your answer

Type x² as x^2.

Squared brackets

$(x + 4)^{2}$ means $(x + 4) (x + 4)$ . Write it out as two brackets before expanding.

The result always has a middle term: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ . Squaring each term separately is the most common mistake in the topic.

1Write as

(x - 7) (x - 7)

x^{2} - 7 x - 7 x + 49

Answer

x^{2} - 14 x + 49

Your turnSame method, new numbers — try it before moving on.

Expand and simplify $(x + 6)^{2}$ .

Your answer

Check by substituting

Test with numbers: if $x = 1$ , then $(x + 6)^{2} = 49$ . The wrong answer $x^{2} + 36$ gives 37, but $x^{2} + 12 x + 36$ gives 49 ✓. Substituting a simple value is a fast check on any expansion.

Three brackets (Higher)

Expand any two brackets first, simplify, then multiply the result by the third bracket. Each of the three terms multiplies each of the two terms, giving up to six products before collecting.

(x + 1) (x + 2) = x^{2} + 3 x + 2

2Multiply by

(x - 4)

x^{3} - 4 x^{2} + 3 x^{2} - 12 x + 2 x - 8

3Collect:

x^{3} - x^{2} - 10 x - 8

Answer

x^{3} - x^{2} - 10 x - 8

Your turnSame method, new numbers — try it before moving on.

Expand and simplify $(x + 3) (x - 1) (x + 2)$ .

Your answer

Type x³ as x^3.

Think like an examiner

Show all four products for double brackets — a correct grid can earn a method mark even if collecting goes wrong.
"Expand and simplify" needs like terms collected; leaving $+ 3 x + 2 x$ loses the final mark.
"Show that" questions need every step written: you must arrive at exactly the given expression.

Expansions

a (b + c) = ab + a c

Every term inside is multiplied.

(a + b) (c + d) = a c + a d + b c + b d

Four products, then collect.

(a + b)^{2} = a^{2} + 2 ab + b^{2}

There is always a middle term.

Watch out for these

3 (x + 4) = 3 x + 4

Multiply both terms:

3 x + 12

- 2 (x - 3) = - 2 x - 6

Negative × negative = positive:

- 2 x + 6

(x + 5)^{2} = x^{2} + 25

Write two brackets:

x^{2} + 10 x + 25

(x + 2) (x + 3) = x^{2} + 6

Include the outer and inner products:

x^{2} + 5 x + 6

Stretch yourself

Show that $(n + 3)^{2} - (n - 3)^{2}$ is always a multiple of 12 for any integer $n$ .

Hint — Expand both squared brackets carefully, then subtract — the bracket around the second one matters.

Questions students ask

Is FOIL or the grid method better?

Both give the same four products. The grid is harder to get wrong and extends naturally to brackets with three terms; FOIL is quicker once you are confident.

Why does

(x + 3)^{2}

not equal

x^{2} + 9

Because squaring means multiplying the whole bracket by itself. The cross terms

3 x + 3 x

come from

x

multiplying 3 and 3 multiplying

x

How do I know I have expanded correctly?

Substitute a simple value such as

x = 2

into the original and your answer. If they give different numbers, there is a mistake.

Key takeaways

Multiply every term inside by the term outside, signs included.
Expand each bracket, then collect like terms.
Double brackets give four products — use a grid or FOIL.
A squared bracket is two brackets: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ .
For three brackets, expand two first, then multiply by the third.

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 23 marks

(a)Expand $5 (2 m - 3)$ .[1]
(b)Expand and simplify $4 (x + 2) + 3 (2 x - 1)$ .[2]

Question 2Grade 53 marks

Show that $(2 x - 3)^{2} - 4 (x - 1) \equiv 4 x^{2} - 16 x + 13$ .[3]

Question 3Grade 73 marks

Expand and simplify $(x - 2) (x + 5) (2 x + 1)$ .[3]

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Expanding Brackets

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Three brackets (Higher)

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Expanding Brackets

Single brackets

Double brackets

Squared brackets

Three brackets (Higher)

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Skill drill

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