GCSE Maths Expanding & Factorising Notes
Expanding multiplies out brackets; factorising is the reverse — writing an expression as a product of brackets. Both are core algebra skills tested across the paper.
Key concepts
- Expanding a single bracket
- Multiply every term inside the bracket by the term outside, e.g. 3(x + 4) = 3x + 12.
- Expanding double brackets
- Multiply each term in the first bracket by each term in the second (FOIL), then collect like terms.
- Factorising (common factor)
- Take out the highest common factor of every term, e.g. 6x + 9 = 3(2x + 3).
- Factorising quadratics
- Find two numbers that multiply to the constant and add to the middle coefficient.
Key formulas
Double brackets
(x + a)(x + b) = x² + (a + b)x + ab
Difference of two squares
a² − b² = (a + b)(a − b)
Worked example
Factorise x² + 7x + 12.
- 1Find two numbers that multiply to 12 and add to 7.
- 23 and 4 work: 3 × 4 = 12 and 3 + 4 = 7.
- 3Write the brackets: (x + 3)(x + 4).
- 4Check by expanding: x² + 4x + 3x + 12 = x² + 7x + 12. ✓
Answer: (x + 3)(x + 4)
Common mistakes
✗ Only multiplying the first term inside the bracket.
✓ Every term inside must be multiplied by the term outside.
✗ Sign errors with negative terms, e.g. (x − 3)(x + 2).
✓ Multiply signs carefully: −3 × +2 = −6, so the constant is −6.
✗ Not taking out the highest common factor.
✓ 6x + 12 = 6(x + 2), not 2(3x + 6) or 3(2x + 4).
Quick check
Try these, then reveal the answers.
Expand 4(2x − 5).Reveal answer
8x − 20
Factorise 5x + 15.Reveal answer
5(x + 3)
Factorise x² − 9.Reveal answer
(x + 3)(x − 3)
Exam tips
- Always check a factorised answer by expanding it back.
- Spot the difference of two squares — it has no middle term.
- Factorise fully: take out numbers and letters that are common to every term.
Related practice
Practise this topic
Exam-style questions with instant Gradion feedback.
Flashcards
Quick-fire recall to lock in the key facts.
Worksheets
Printable practice you can work through offline.
Past papers
Full papers with examiner-grade marking.
Gradion
Ask for a step-by-step explanation any time.
All Mathematics
Every Mathematics topic across every exam board.
Related topics
Frequently asked questions
What is the difference between expanding and factorising?+
Expanding removes brackets by multiplying out; factorising puts brackets back by writing the expression as a product.
How do I factorise a quadratic?+
Find two numbers that multiply to the constant term and add to the coefficient of x, then write them in two brackets.
Create a free account to practise expanding & factorising with Gradion feedback
Try 5 questions now, get instant examiner-grade marking and ask the Gradion to explain anything. Free during our early launch period — no credit card required.