1MA1 · A4AlgebraFoundation & HigherGrades 2–7

Expanding Brackets

Q: What is the quickest way to expand (x - 6)^2?

Use a^2 - 2ab + b^2: x^2 - 12x + 36.

Expanding brackets removes them by multiplying out. It is one rule — every term inside is multiplied by every term outside — applied with increasing care as brackets multiply: one bracket, two brackets, then three. Most errors come from signs and from missing a pair of terms, so a systematic layout matters more than speed.

50 min Algebraic Expressions

The big picture

Expanding is the reverse of factorising, and both rest on the distributive law: $a (b + c) = ab + a c$ . A grid, or the discipline of pairing each term in turn, guarantees that no product is missed.

Edexcel questions go beyond routine expansion: "show that" an area or volume simplifies to a given expression, prove an identity, or find unknown coefficients such as $a$ and $b$ in $(x + a) (x + b) \equiv x^{2} + 7 x + 12$ . All of them need clean, reliable expansion.

What you'll learn

Expand single brackets and simplify, including with negative terms outside
Expand the product of two binomials
Expand squared brackets and recognise the difference of two squares
Expand the product of three binomials (Higher)

Single brackets

Multiply each term inside by the term outside: $4 (2 x - 5) = 8 x - 20$ .

A negative outside changes every sign inside: $- 3 (x - 4) = - 3 x + 12$ .

Expand each bracket first, then collect like terms: $5 (x + 2) - 2 (3 x - 1) = 5 x + 10 - 6 x + 2 = - x + 12$ .

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

Expand $- 4 (2 y - 5)$ .

Your answer

Tip — Treat the sign in front of a bracket as part of the multiplier. In $- 2 (3 x - 1)$ the multiplier is $- 2$ , and $- 2 \times - 1 = + 2$ .

Two binomials

For $(x + 3) (x - 7)$ , multiply each term in the first bracket by each term in the second: four products in total.

A grid keeps it organised: $x \times x = x^{2}$ , $x \times (- 7) = - 7 x$ , $3 \times x = 3 x$ , $3 \times (- 7) = - 21$ .

Collect the middle terms: $x^{2} - 4 x - 21$ .

3 y \times 2 y = 6 y^{2}

;

3 y \times 5 = 15 y

- 2 \times 2 y = - 4 y

;

- 2 \times 5 = - 10

3Collect:

6 y^{2} + 11 y - 10

Answer

6 y^{2} + 11 y - 10

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

Expand and simplify $(4 x - 3) (x + 5)$ .

Your answer

Type powers with ^, e.g. 3x^2 - 4x.

Squared brackets and special products

$(a + b)^{2}$ means $(a + b) (a + b)$ , which gives $a^{2} + 2 ab + b^{2}$ — the middle term is doubled.

$(a - b)^{2} = a^{2} - 2 ab + b^{2}$ .

$(a + b) (a - b) = a^{2} - b^{2}$ : the middle terms cancel, giving the $difference of two squares$ .

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

Squared bracket.

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

Difference of two squares.

1(a)

4 x^{2} - 2 (2 x) (3) + 9 = 4 x^{2} - 12 x + 9

2(b) Difference of two squares:

25 m^{2} - 16 n^{2}

Answer(a)

4 x^{2} - 12 x + 9

(b)

25 m^{2} - 16 n^{2}

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

Expand and simplify $(3 a + 2)^{2}$ .

Your answer

Type powers with ^, e.g. 3x^2 - 4x.

Why this works

$(x + 5)^{2}$ is not $x^{2} + 25$ . At $x = 1$ , the true value is $36$ while $x^{2} + 25$ gives $26$ — the missing $10 x$ is exactly the difference.

Three binomials

Expand two of the brackets first, simplify, then multiply the result by the third bracket.

The final multiplication produces up to six terms before simplifying, so write them all out before collecting.

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

2Multiply by

(2 x + 3)

2 x^{3} + 3 x^{2} - 2 x^{2} - 3 x - 4 x - 6

3Collect:

2 x^{3} + x^{2} - 7 x - 6

AnswerBoth sides are identical, as required.

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

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

Your answer

Type powers with ^, e.g. 3x^2 - 4x.

Tip — In "show that" questions every step must be visible. Write the unsimplified six-term line before collecting like terms.

Think like an examiner

Show all products before collecting like terms.
For identities, expand and then state that both sides match.
When finding unknown coefficients, expand and compare terms explicitly.

Expansion

a (b + c) = ab + a c

Distributive law.

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

Two binomials.

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

Squared difference.

Watch out for these

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

- 2 \times - 3 = + 6

, so

- 2 x + 6

(x + 4)^{2} = x^{2} + 16

Include the middle term:

x^{2} + 8 x + 16

Multiplying only first and last terms in two brackets.

There are four products: every term by every term.

(3 x)^{2}

written as

3 x^{2}

inside an expansion.

(3 x)^{2} = 9 x^{2}

Stretch yourself

Given that $(2 x + a) (x - 3) \equiv 2 x^{2} + b x - 15$ , find the values of $a$ and $b$ .

Hint — Expand the left side and compare the constant terms first.

Questions students ask

Is FOIL the same as the grid method?

Yes — both produce the same four products for two brackets. The grid extends more easily to longer brackets.

Do I always need to simplify?

Yes, unless the question asks otherwise. "Expand and simplify" means collect like terms.

What is the quickest way to expand

(x - 6)^{2}

Use

a^{2} - 2 ab + b^{2}

x^{2} - 12 x + 36

Key takeaways

Multiply every term inside by every term outside.
A negative multiplier changes every sign inside the bracket.
Two binomials give four products; collect the middle terms.
$(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}$ and $(a + b) (a - b) = a^{2} - b^{2}$ .
For three brackets, expand two first, then multiply by the third.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 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 32 marks

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

Question 2Grade 52 marks

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

Question 3Grade 73 marks

Show that $(x + 4) (x - 3) (2 x - 1) \equiv 2 x^{3} + x^{2} - 25 x + 12$ .[3]

Unlimited practice

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Expanding Brackets

Single brackets

Two binomials

Squared brackets and special products

Three binomials

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Expanding Brackets

Single brackets

Two binomials

Squared brackets and special products

Three binomials

Practise it the way it is examined

Skill drill

Practise this topic

Related topics