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Expanding Brackets

FoundationHigherAQAEdexcelOCR

Expanding Brackets is a key algebra topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on expanding single and double brackets, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Multiply every inside term by the outside term; use FOIL for double brackets.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Expanding brackets means multiplying everything inside by the term outside, removing the brackets in the process. The term outside multiplies each term inside, one at a time.

So \(3(x + 4)\) becomes \(3x + 12\), because both the \(x\) and the \(4\) are multiplied by \(3\). Multiplying only the first term is the most common error and loses the mark immediately.

A negative outside the bracket changes the sign of every term inside. In \(-2(x - 5)\), the result is \(-2x + 10\), because a negative times a negative gives a positive. Writing the signs carefully before simplifying prevents most slips.

Revision notes

Single brackets

Multiply each term inside by the term outside.

For \(4(2x - 3)\): \(4 \times 2x = 8x\) and \(4 \times -3 = -12\), giving \(8x - 12\).

Negative multipliers

A negative outside flips the sign of every term inside.

For \(-3(x - 2)\): \(-3 \times x = -3x\) and \(-3 \times -2 = +6\), giving \(-3x + 6\). The second sign changes because two negatives make a positive.

Expanding and simplifying

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

For \(2(x + 3) + 4(x - 1)\): expanding gives \(2x + 6 + 4x - 4\), which simplifies to \(6x + 2\). Expand before collecting, never the other way round.

Key points

  • Multiply every term inside by the term outside.
  • A negative outside changes all the signs inside.
  • Two negatives multiply to give a positive.
  • Expand fully before collecting like terms.
  • The number of terms inside determines the number of products.
  • Check by substituting a value into both forms.

Worked examples

Example 1

Expand \(5(3x - 2)\).

Working

\[5 \times 3x = 15x\]multiply the first term
\[5 \times -2 = -10\]multiply the second term
\[15x - 10\]write the expanded expression

Example 2

Expand \(-4(2x + 3)\).

Working

\[-4 \times 2x = -8x\]multiply the first term, keeping the negative
\[-4 \times 3 = -12\]multiply the second term
\[-8x - 12\]write the expanded expression

Example 3

Expand and simplify \(3(x + 5) - 2(x - 1)\).

Working

\[3x + 15\]expand the first bracket
\[-2x + 2\]expand the second, noting −2 × −1 = +2
\[x + 17\]collect like terms

Common mistakes

  • Multiplying only the first term inside.

    3(x + 4) is 3x + 12, not 3x + 4. Every term inside is multiplied.

  • Getting the sign wrong after a negative bracket.

    −2(x − 5) gives −2x + 10. The second sign flips.

  • Collecting before expanding.

    Brackets must be removed first, or the terms inside are not yet available to collect.

  • Forgetting the invisible 1.

    −(x − 3) means −1(x − 3), which gives −x + 3.

Exam tips

  • Draw arrows from the outside term to each term inside.
  • Write every sign in before simplifying.
  • Expand each bracket on its own line for clarity.
  • Substitute x = 2 into both forms to check they agree.

Key terms

Expand
To multiply out brackets.
Bracket
A grouping symbol showing which terms are multiplied together.
Distribute
To multiply the outside term across every term inside.
Simplify
To collect like terms after expanding.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.