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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn8300Papers 1–3 · Higher tierAlgebraExpanding and factorising
Algebra · Subtopic 1 of 11

Expanding and Factorising

Expanding removes brackets; factorising puts them back. They are the same skill in reverse: every term in one bracket multiplies every term in the other, and factorising finds the brackets that would multiply out to give the expression.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27
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Section 1 of 14

Learning objectives

  • Expand single and double brackets, and squared brackets, and simplify.
  • Factorise by taking out the highest common factor, and by grouping.
  • Factorise quadratics x2+bx+cx^2+bx+c and ax2+bx+cax^2+bx+c.
  • Recognise and factorise the difference of two squares.
Pause and think

Work out 23×1723\times17 in your head by thinking of it as (20+3)(20−3)(20+3)(20-3). What pattern makes this easy?

Section 2 of 14

Core idea

Start hereArea of a rectangle, two ways

A rectangle x+3x+3 by x+2x+2 can be cut into four smaller rectangles: x2x^2, 2x2x, 3x3x, 66. So (x+3)(x+2)=x2+5x+6(x+3)(x+2)=x^2+5x+6. Expanding is just adding up pieces of area; factorising is reassembling them.

The big idea

Expand: multiply every term by every term. Factorise: common factor first, then patterns: (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2, a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b).

1. Expanding

(x + 3)(x + 2) as the area of a rectangle
x²3x2x6x3x2
Four pieces: x2+3x+2x+6=x2+5x+6x^2+3x+2x+6=x^2+5x+6.
Worked exampleExpand

3(2x−5)3(2x-5); (x+3)(x+2)(x+3)(x+2); (2x−1)2(2x-1)^2.

Goal
Expanded form.
Strategy
Each term times each term.
  1. 6x−156x-15
    33 times both.
  2. x2+2x+3x+6=x2+5x+6x^2+2x+3x+6=x^2+5x+6
    Four products, then collect.
  3. (2x−1)(2x−1)=4x2−4x+1(2x-1)(2x-1)=4x^2-4x+1
    Write the square as two brackets.
Answer6x−156x-15; x2+5x+6x^2+5x+6; 4x2−4x+14x^2-4x+1.
✓ Checkx=1x=1: (2−1)2=1(2-1)^2=1 and 4−4+1=14-4+1=1 ✓.
Common misconception(2x−1)2≠4x2+1(2x-1)^2\neq4x^2+1 The middle term −4x-4x comes from the two cross products.

2. Factorising

Worked exampleFour types

6x2+9x6x^2+9x; x2−x−12x^2-x-12; 2x2+7x+32x^2+7x+3; 9x2−259x^2-25.

Goal
Factorised.
Strategy
Common factor; sum-product; split the middle; difference of squares.
  1. 3x(2x+3)3x(2x+3)
    HCF 3x3x.
  2. (x−4)(x+3)(x-4)(x+3)
    Multiply to −12-12, add to −1-1.
  3. 2x2+6x+x+3=2x(x+3)+1(x+3)=(2x+1)(x+3)2x^2+6x+x+3=2x(x+3)+1(x+3)=(2x+1)(x+3)
    ac=6ac=6: split 7x7x as 6x+x6x+x.
  4. (3x−5)(3x+5)(3x-5)(3x+5)
    (3x)2−52(3x)^2-5^2.
AnswerAs shown.
✓ CheckExpand (2x+1)(x+3)=2x2+7x+3(2x+1)(x+3)=2x^2+7x+3 ✓.

3. Algebraic fractions

Worked exampleSimplify x2−4x2+x−2\frac{x^2-4}{x^2+x-2}

Simplify.

Goal
Lowest terms.
Strategy
Factorise top and bottom; cancel common factors.
  1. (x−2)(x+2)(x+2)(x−1)\frac{(x-2)(x+2)}{(x+2)(x-1)}
    Factorise.
  2. x−2x−1\frac{x-2}{x-1}
    Cancel (x+2)(x+2).
Answerx−2x−1\frac{x-2}{x-1}.
✓ Checkx=3x=3: 510=12\frac5{10}=\frac12 ✓.
Worked example2x+3x+1\frac2x+\frac3{x+1}

Single fraction.

Goal
Combined.
Strategy
Common denominator x(x+1)x(x+1).
  1. 2(x+1)x(x+1)+3xx(x+1)\frac{2(x+1)}{x(x+1)}+\frac{3x}{x(x+1)}
    Scale each.
  2. 5x+2x(x+1)\frac{5x+2}{x(x+1)}
    Add tops.
Answer5x+2x(x+1)\frac{5x+2}{x(x+1)}.
✓ Checkx=1x=1: 2+1.5=3.5=722+1.5=3.5=\frac72 ✓.
Common misconceptionCancelling terms In x+2x\frac{x+2}{x} you can't cancel the xx's: only whole factors cancel.
Pause and think

Factorise x2−10x+25x^2-10x+25.

Summary

  • Expand every pair.
  • Factorise: common factor first.
  • Cancel only factors.
Key words
Expand
Multiply out brackets.
Factorise
Write as a product.
HistoryRestoring and balancing. The word "algebra" comes from al-jabr, in the title of a book written around 820820 by Muhammad ibn Musa al-Khwarizmi in Baghdad's House of Wisdom. Al-jabr means "restoring", moving a subtracted term to the other side of an equation. His name gives us the word "algorithm".
Fun factBone-setters. Al-jabr also meant setting broken bones, and in medieval Spain an algebrista was a person who did exactly that, often also a barber. Restoring a bone and restoring an equation used the same word.
Section 3 of 14

Key formulae

Double brackets(x+a)(x+b)=x2+(a+b)x+ab(x+a)(x+b)=x^2+(a+b)x+ab
Perfect square(x+a)2=x2+2ax+a2(x+a)^2=x^2+2ax+a^2
Difference of two squaresa2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b)
TrickThe grid method. Draw a 2×22\times2 grid with xx and 55 along the top and xx and −3-3 down the side. Fill each box with the product (x2x^2, 5x5x, −3x-3x, −15-15) and add them up. No term gets forgotten, and the negatives stay visible.
Section 4 of 14

Worked examples

Example 1 · Factorising a quadratic0 / 2 steps
Factorise x2−7x+12x^2-7x+12.
    Example 2 · When a is not 10 / 2 steps
    Factorise 2x2+5x−32x^2+5x-3.
      Example 3 · Difference of two squares0 / 2 steps
      Factorise 4x2−254x^2-25.
        Section 5 of 14

        Explore the graph

        Factorised and expanded forms are the same curve
        Loading the interactive graph…
        Try this
        1. (x+a)(x+b)(x+a)(x+b) and x2+(a+b)x+abx^2+(a+b)x+ab draw exactly the same curve, whatever aa and bb are.
        2. The curve crosses the xx-axis at x=−ax=-a and x=−bx=-b. Why does the factorised form show this immediately?
        3. Make a=−ba=-b. Which special factorisation do you get?

        Factorising and expanding rewrite an expression without changing its value.

        In the real worldMental maths shortcuts. 992=(100−1)2=10 000−200+1=980199^2=(100-1)^2=10\,000-200+1=9801, and 52×48=502−22=249652\times48=50^2-2^2=2496. Expanding brackets is how people who seem to multiply instantly actually do it.
        Section 6 of 14

        Common mistakes

        Incorrect(x+3)2=x2+9(x+3)^2=x^2+9
        Correctx2+6x+9x^2+6x+9
        Square a bracket by multiplying it by itself.
        Incorrect−2(x−1)=−2x−2-2(x-1)=-2x-2
        Correct−2x+2-2x+2
        Negative times negative is positive.
        Incorrect6x2+9x=x(6x+9)6x^2+9x=x(6x+9)
        Correct3x(2x+3)3x(2x+3)
        Take out the highest common factor.
        Incorrectx2−16=(x−4)2x^2-16=(x-4)^2
        Correct(x−4)(x+4)(x-4)(x+4)
        Difference of two squares has one ++ and one −-.

        (x+3)2=x2+9(x+3)^2=x^2+9

        (x+3)(x+3)(x+3)(x+3), every term times every term

        Section 7 of 14

        Exam tip

        Exam tipExpand your answer to check it. After factorising, multiply your brackets back out in your head. If you don't get the original expression, something is wrong. It takes ten seconds and catches nearly every sign error.
        Section 8 of 14

        Guided practice

        SA-4024-AL1-0012 marksL1 FoundationCalculatorExpanding
        Expand and simplify 3(x+4)−2(x−1)3(x+4)-2(x-1).

        Sign in or create a free account to check answers, get hints and track your progress.

        Salam Academy Original Practice · Proposed mark scheme
        Section 9 of 14

        Practice questions

        SA-4024-AL1-0022 marksL1 FoundationCalculatorExpanding
        Expand and simplify (x+5)(x−3)(x+5)(x-3).

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL5-0012 marksL1 FoundationNon-calculatorExpanding
        (2x−3)2=4x2+bx+c(2x-3)^2=4x^2+bx+c. Find bb and cc.

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL5-0022 marksL1 FoundationNon-calculatorFactorising
        12xy−8x212xy-8x^2 factorises completely as kx(3y−2x)kx(3y-2x). Find kk.

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL1-0051 markL1 FoundationCalculatorFactorising
        6x2+9x6x^2+9x factorises completely as 3x(2x+k)3x(2x+k). Find kk.

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL5-0032 marksL2 StandardNon-calculatorFactorising
        x2+2x−15=(x+a)(x+b)x^2+2x-15=(x+a)(x+b) with a>ba>b. Find aa and bb.

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL1-0032 marksL1 FoundationCalculatorFactorising
        x2−7x+12=(x−a)(x−b)x^2-7x+12=(x-a)(x-b) with a<ba<b. Find aa and bb.

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL5-0042 marksL2 StandardNon-calculatorFactorising
        Factorise 3x+3y+ax+ay3x+3y+ax+ay by grouping. Which is correct?
        Choose one option

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL1-0061 markL2 StandardCalculatorFactorising
        4x2−25=(2x−a)(2x+a)4x^2-25=(2x-a)(2x+a). Find aa.

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL5-0051 markL2 StandardNon-calculatorExpanding
        Use a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b) to work out 52×4852\times48 without a calculator.

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        Salam Academy Original Practice · Proposed mark scheme
        SA-4024-AL-0013 marksL2 StandardNon-calculatorExpanding
        Expand and simplify (2x+3)(x−4)−2x(x−1)(2x+3)(x-4)-2x(x-1).

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        Salam Academy Original Practice · Proposed mark scheme

        More questions on every skill in this subtopic are in the question bank under Practise.

        Section 10 of 14

        Challenge

        SA-4024-AL-0023 marksL2 StandardNon-calculatorFactorising
        Factorise 6x2+7x−36x^2+7x-3, and hence solve 6x2+7x−3=06x^2+7x-3=0.

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        Salam Academy Original Practice · Proposed mark scheme
        Section 11 of 14

        Summary

        • Expand: every term times every term, then collect like terms.
        • Factorise: HCF first, then group, quadratic or difference of two squares.
        • x2+bx+cx^2+bx+c: two numbers multiplying to cc and adding to bb.
        • a2−b2=(a−b)(a+b)a^2-b^2=(a-b)(a+b).
        Section 12 of 14

        Quick quiz

        Q1 · SA-4024-AL-Q011 markL1 FoundationCalculatorFactorising
        x2−16x^2-16 factorises to
        Choose one option

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        Salam Academy Original Practice · Proposed mark scheme
        Section 13 of 14

        Same topic, different board

        Specification
        GCSE Mathematics 8300
        Where
        Algebra
        Expected
        The same; expanding three brackets is also tested at Higher.
        Calculator
        Paper 1 non-calculator; Papers 2 and 3 calculator.
        Typical style
        Problem-solving questions often combine topics.
        Open in
        8300 · Papers 1–38300 · Papers 1–3
        Section 14 of 14

        Past paper questions

        6 questions (17 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

        1. 4024/22 · Oct/Nov 2025Question 72 marks

          Factorise an expression.

          Question paper, page 5Mark scheme
        2. 4024/22 · Oct/Nov 2025Question 82 marks

          Expand and simplify brackets.

          Question paper, page 5Mark scheme
        3. 4024/22 · Oct/Nov 2025Question 105 marks

          Combine and rearrange two algebraic expressions.

          Question paper, page 6Mark scheme
        4. 4024/22 · May/June 2025Question 122 marks

          Factorise an expression.

          Question paper, page 8Mark scheme
        5. 4024/12 · Oct/Nov 2024Question 112 marks

          Factorising.

          Question paper, page 7Mark scheme
        6. 4024/12 · Oct/Nov 2024Question 204 marks

          Laws of indices and expanding brackets.

          Question paper, page 13Mark scheme

        All 4024 past paper questions by topic

        Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 8300, First teaching 2015 (official specification).