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, $3x$ and $12$ . Both describe the same area, so $3(x + 4) \equiv 3x + 12$ .

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.

Open on YouTube

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: $a(b + c) = ab + ac$ .

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

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

Example 1Expand

4(3x - 5)

4 \times 3x = 12x

4 \times (-5) = -20

Answer

12x - 20

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

Expand $6(2y + 7)$ .

Your answer

Show worked solution

Solution

1 $6 \times 2y = 12y$ .
2 $6 \times 7 = 42$ .

Answer: $12y + 42$

Example 2Expand and simplify

3(2a + 1) - 2(a - 4)

3(2a + 1) = 6a + 3

-2(a - 4) = -2a + 8

3Collect:

6a - 2a + 3 + 8

Answer

4a + 11

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

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

Your answer

Show hint

The $-3$ multiplies both terms: $-3 \times -2 = +6$ .

Show worked solution

Solution

1 $5(x + 3) = 5x + 15$ .
2 $-3(x - 2) = -3x + 6$ .
3Collect: $2x + 21$ .

Answer: $2x + 21$

Example 3Expand

2x(x - 7)

2x \times x = 2x^2

2x \times (-7) = -14x

Answer

2x^2 - 14x

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 + 3x + 2x + 6 = x^2 + 5x + 6

Example 1Expand and simplify

(x + 5)(x - 3)

1First:

x \times x = x^2

. Outer:

x \times (-3) = -3x

2Inner:

5 \times x = 5x

. Last:

5 \times (-3) = -15

3Collect the

x

terms:

-3x + 5x = 2x

Answer

x^2 + 2x - 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.

Show worked solution

Solution

1 $x^2 - 6x - 4x + 24$ .
2Collect: $x^2 - 10x + 24$ .

Answer: $x^2 - 10x + 24$

Example 2Expand and simplify

(2x + 3)(4x - 1)

2x \times 4x = 8x^2

and

2x \times (-1) = -2x

3 \times 4x = 12x

and

3 \times (-1) = -3

3Collect:

8x^2 + 10x - 3

Answer

8x^2 + 10x - 3

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

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

Your answer

Type x² as x^2.

Show worked solution

Solution

1 $6x^2 + 15x - 4x - 10$ .
2Collect: $6x^2 + 11x - 10$ .

Answer: $6x^2 + 11x - 10$

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: $(a + b)^2 = a^2 + 2ab + b^2$ . Squaring each term separately is the most common mistake in the topic.

Example 1Expand and simplify

(x - 7)^2

1Write as

(x - 7)(x - 7)

x^2 - 7x - 7x + 49

Answer

x^2 - 14x + 49

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

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

Your answer

Show hint

Write it as $(x + 6)(x + 6)$ — there should be four products.

Show worked solution

Solution

1 $(x + 6)(x + 6) = x^2 + 6x + 6x + 36$ .
2Collect: $x^2 + 12x + 36$ .

Answer: $x^2 + 12x + 36$

Check by substituting

Test with numbers: if $x = 1$ , then $(x + 6)^2 = 49$ . The wrong answer $x^2 + 36$ gives 37, but $x^2 + 12x + 36$ 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.

Example 1Expand and simplify

(x + 1)(x + 2)(x - 4)

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

2Multiply by

(x - 4)

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

3Collect:

x^3 - x^2 - 10x - 8

Answer

x^3 - x^2 - 10x - 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.

Show worked solution

Solution

1 $(x + 3)(x - 1) = x^2 + 2x - 3$ .
2Multiply by $(x + 2)$ : $x^3 + 2x^2 + 2x^2 + 4x - 3x - 6$ .
3Collect: $x^3 + 4x^2 + x - 6$ .

Answer: $x^3 + 4x^2 + x - 6$

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 $+3x + 2x$ loses the final mark.
"Show that" questions need every step written: you must arrive at exactly the given expression.

Expansions

a(b + c) = ab + ac

Every term inside is multiplied.

(a + b)(c + d) = ac + ad + bc + bd

Four products, then collect.

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

There is always a middle term.

Watch out for these

3(x + 4) = 3x + 4

Multiply both terms:

3x + 12

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

Negative × negative = positive:

-2x + 6

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

Write two brackets:

x^2 + 10x + 25

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

Include the outer and inner products:

x^2 + 5x + 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.

Reveal solution

Solution

(n + 3)^2 = n^2 + 6n + 9

and

(n - 3)^2 = n^2 - 6n + 9

. Subtracting:

(n^2 + 6n + 9) - (n^2 - 6n + 9) = 12n

. Since

n

is an integer,

12n

is a multiple of 12.

\blacksquare

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

3x + 3x

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: $(a + b)^2 = a^2 + 2ab + b^2$ .
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 $5(2m - 3)$ .[1]
Show mark scheme (a)
- B1 $10m - 15$
Answer: $10m - 15$
(b)Expand and simplify $4(x + 2) + 3(2x - 1)$ .[2]
Show mark scheme (b)
- M1Three of the four terms correct in $4x + 8 + 6x - 3$
- A1 $10x + 5$
Answer: $10x + 5$

Question 2Grade 53 marks

Show that $(2x - 3)^2 - 4(x - 1) \equiv 4x^2 - 16x + 13$ .[3]
Show mark scheme
- M1Expands $(2x - 3)^2$ with four terms, at least three correct: $4x^2 - 6x - 6x + 9$
- M1Expands $-4(x - 1)$ correctly: $-4x + 4$
- A1Collects to $4x^2 - 16x + 13$ with no errors seen

Examiner note

In "show that" questions the answer is given, so all the marks are for the working. Write both expansions in full before collecting.

Question 3Grade 73 marks

Expand and simplify $(x - 2)(x + 5)(2x + 1)$ .[3]
Show mark scheme
- M1Expands two brackets correctly, e.g. $(x - 2)(x + 5) = x^2 + 3x - 10$
- M1Multiplies their quadratic by the third bracket, with at least 5 of 6 terms correct
- A1 $2x^3 + 7x^2 - 17x - 10$
Answer: $2x^3 + 7x^2 - 17x - 10$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Expand $3(x + 4)$ .
- $3x + 12$
- $3x + 4$
- $x + 12$
- $7x$
Show answer and explanation
Answer: $3x + 12$
Multiply both terms by 3.
Expand $-2(y - 5)$ .
- $-2y + 10$
- $-2y - 10$
- $2y - 10$
- $-2y - 5$
Show answer and explanation
Answer: $-2y + 10$
$-2 \times -5 = +10$ .
Expand $a(a + 6)$ .
- $a^2 + 6a$
- $2a + 6$
- $a^2 + 6$
- $7a$
Show answer and explanation
Answer: $a^2 + 6a$
$a \times a = a^2$ and $a \times 6 = 6a$ .
Expand and simplify $2(x + 3) + 4(x - 1)$ .
- $6x + 2$
- $6x + 10$
- $6x - 2$
- $8x + 2$
Show answer and explanation
Answer: $6x + 2$
$2x + 6 + 4x - 4$ .
Expand and simplify $(x + 2)(x + 7)$ .
- $x^2 + 9x + 14$
- $x^2 + 14$
- $x^2 + 9x + 9$
- $2x + 9$
Show answer and explanation
Answer: $x^2 + 9x + 14$
$x^2 + 7x + 2x + 14$ .
Expand and simplify $(x - 3)^2$ .
- $x^2 - 6x + 9$
- $x^2 + 9$
- $x^2 - 9$
- $x^2 - 6x - 9$
Show answer and explanation
Answer: $x^2 - 6x + 9$
$(x - 3)(x - 3) = x^2 - 3x - 3x + 9$ .
Expand and simplify $(3x + 1)(x - 2)$ .
- $3x^2 - 5x - 2$
- $3x^2 - 7x - 2$
- $3x^2 - 2$
- $3x^2 + 5x - 2$
Show answer and explanation
Answer: $3x^2 - 5x - 2$
$3x^2 - 6x + x - 2$ .
What is the coefficient of $x$ in $(x + 1)(x + 2)(x + 3)$ ?
- $11$
- $6$
- $5$
- $3$
Show answer and explanation
Answer: $11$
$(x^2 + 3x + 2)(x + 3) = x^3 + 6x^2 + 11x + 6$ .

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