Expanding Brackets and Factorising

Expanding brackets and factorising are two of the most important algebraic skills in AQA GCSE Mathematics. They are essentially opposite operations: expanding removes brackets from an expression, while factorising puts them in. Mastering both skills is essential for solving equations, simplifying expressions, and tackling quadratics at every grade level.

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Expanding a Single Bracket

To expand a single bracket, multiply every term inside the bracket by the term outside.

Rule

a(b + c) = ab + ac

Worked Example 1

Expand: 3(2x + 5)

Multiply each term inside the bracket by 3:

• 3 x 2x = 6x
• 3 x 5 = 15

Answer: 6x + 15

Worked Example 2

Expand: -2(4y - 3)

Multiply each term inside the bracket by -2:

• -2 x 4y = -8y
• -2 x (-3) = +6

Answer: -8y + 6

Exam Tip: Take extra care with negative signs when expanding. The most common error is forgetting that a negative multiplied by a negative gives a positive.

Worked Example 3

Expand and simplify: 3(x + 4) + 2(x - 1)

Expand each bracket:

• 3(x + 4) = 3x + 12
• 2(x - 1) = 2x - 2

Collect like terms: 3x + 2x + 12 - 2 = 5x + 10

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Expanding Double Brackets

When multiplying two brackets together, every term in the first bracket must be multiplied by every term in the second bracket. A helpful method is FOIL: First, Outside, Inside, Last.

Rule

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

flowchart LR
    A["(a + b)(c + d)"] --> B[First: a x c]
    A --> C[Outside: a x d]
    A --> D[Inside: b x c]
    A --> E[Last: b x d]
    B --> F[ac + ad + bc + bd]
    C --> F
    D --> F
    E --> F
    F --> G[Collect like terms]

Worked Example 4

Expand: (x + 3)(x + 5)

• First: x x x = x^2
• Outside: x x 5 = 5x
• Inside: 3 x x = 3x
• Last: 3 x 5 = 15

Combine: x^2 + 5x + 3x + 15 = x^2 + 8x + 15

Worked Example 5

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

• First: 2x x x = 2x^2
• Outside: 2x x 4 = 8x
• Inside: -1 x x = -x
• Last: -1 x 4 = -4

Combine: 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4

Expanding a Bracket Squared

(x + a)^2 means (x + a)(x + a). Do not simply square each term individually.

Worked Example 6

Expand: (x + 3)^2

Write as: (x + 3)(x + 3)

• First: x^2
• Outside: 3x
• Inside: 3x
• Last: 9

Combine: x^2 + 6x + 9

Exam Tip: A very common mistake is to write (x + 3)^2 = x^2 + 9. This is WRONG. You must expand using FOIL. The correct answer is x^2 + 6x + 9 — there is always a middle term.

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Expanding Triple Brackets [H]

For Higher tier, you may need to expand three brackets. The strategy is to expand two brackets first, then multiply the result by the third bracket.

Worked Example 7

Expand: (x + 1)(x + 2)(x + 3)

Step 1 — Expand the first two brackets: (x + 1)(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2

Step 2 — Multiply by the third bracket: (x^2 + 3x + 2)(x + 3)

= x^2 x x + x^2 x 3 + 3x x x + 3x x 3 + 2 x x + 2 x 3

= x^3 + 3x^2 + 3x^2 + 9x + 2x + 6

= x^3 + 6x^2 + 11x + 6

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Factorising into a Single Bracket

Factorising is the reverse of expanding. To factorise an expression into a single bracket, find the highest common factor (HCF) of all terms and place it outside the bracket.

Worked Example 8

Factorise: 6x + 15

HCF of 6 and 15 is 3.

6x / 3 = 2x and 15 / 3 = 5

Answer: 3(2x + 5)

Worked Example 9

Factorise: 4x^2 - 10x

HCF of 4x^2 and 10x is 2x.

4x^2 / 2x = 2x and -10x / 2x = -5

Answer: 2x(2x - 5)

Exam Tip: Always check your factorisation by expanding the bracket back out. If you get the original expression, your factorisation is correct.

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Difference of Two Squares

When two square terms are subtracted, the expression factorises into a special pattern:

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

Worked Example 10

Factorise: x^2 - 25

Recognise that x^2 = (x)^2 and 25 = (5)^2

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

Worked Example 11

Factorise: 4y^2 - 9

Recognise that 4y^2 = (2y)^2 and 9 = (3)^2

Answer: (2y + 3)(2y - 3)

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Factorising Quadratic Expressions

A quadratic expression has the form ax^2 + bx + c. To factorise it into two brackets:

When a = 1 (Simpler Quadratics)

Find two numbers that multiply to give c and add to give b.

Worked Example 12

Factorise: x^2 + 7x + 12

Find two numbers that multiply to 12 and add to 7: 3 and 4.

Answer: (x + 3)(x + 4)

Worked Example 13

Factorise: x^2 - 2x - 15

Find two numbers that multiply to -15 and add to -2: -5 and +3.

Answer: (x - 5)(x + 3)

When a is not equal to 1 (Harder Quadratics) [H]

Use the ac method: multiply a and c, find two numbers that multiply to ac and add to b, then split the middle term and factorise by grouping.

Worked Example 14

Factorise: 2x^2 + 7x + 3

Step 1: a x c = 2 x 3 = 6