Expanding single brackets

Expanding a single bracket means multiplying the term outside the bracket by every term inside it, then writing the answer without brackets. For example, 5(x + 3) = 5 × x + 5 × 3 = 5x + 15. Watch the signs: a negative outside the bracket changes the sign of every term inside. Expanding single brackets is on both GCSE Maths tiers.

How to expand single brackets

1. Multiply the term outside the bracket by the first term inside.
2. Multiply it by the next term inside, keeping track of the sign.
3. Repeat for every term in the bracket.
4. Write the results with their signs, and collect any like terms.

This rule is the distributive law. You can see why it works with areas. A rectangle 4 tall and x + 3 wide splits into two smaller rectangles:

The whole area is 4(x + 3), and the two parts are 4x and 12, so 4(x + 3) = 4x + 12.

Worked examples

Example 1: a simple bracket

Expand 6(x + 2).

6 × x = 6x and 6 × 2 = 12, so 6(x + 2) = 6x + 12.

Example 2: a negative inside

Expand 7(2a − 3).

7 × 2a = 14a and 7 × (−3) = −21, so 7(2a − 3) = 14a − 21.

Example 3: a negative outside

Expand −4(x − 5).

−4 × x = −4x, and −4 × (−5) = +20, because a negative times a negative is positive. So −4(x − 5) = −4x + 20.

Example 4: a letter outside

Expand x(x + 9).

x × x = x² and x × 9 = 9x, so x(x + 9) = x² + 9x.

Example 5: powers

Expand 3y(2y² − y + 4).

3y × 2y² = 6y³, 3y × (−y) = −3y² and 3y × 4 = 12y. So the answer is 6y³ − 3y² + 12y. Multiplying y by y² gives y³: with the same letter, the indices add, 1 + 2 = 3.

Try one yourself in this question from the Algebra course, then press Mark.

Expand and simplify

When there are two brackets added or subtracted, expand each one, then collect like terms.

Example 6: adding two brackets

Example 7: subtracting a bracket

The second bracket is multiplied by −2, not 2, so −2 × (−1) = +2. This is the step people most often get wrong.

Solving equations with brackets

Expanding the bracket is often the first step in solving an equation.

Example 8

Solve 2(5x + 3) = 46.

Check: 2(5 × 4 + 3) = 2 × 23 = 46. Here you could also divide both sides by 2 first, giving 5x + 3 = 23, which leads to the same answer.

Check your understanding.

Practice questions

1. Expand 9(x + 4).
2. Expand −5(2 − 3x).
3. Expand 2p(p + 6).
4. Expand and simplify 3(x + 5) + 4(x − 2).
5. Expand and simplify 6(2x − 1) − 3(x + 2).
6. Solve 4(3x − 2) = 40.

Answers: 1) 9x + 36. 2) −10 + 15x, usually written 15x − 10. 3) 2p² + 12p. 4) 7x + 7. 5) 9x − 12. 6) 12x − 8 = 40, so 12x = 48 and x = 4.

Expanding is the reverse of factorising

Factorising undoes expanding: it puts the brackets back in. 6x + 12 = 6(x + 2), because 6 is a common factor of both terms. Check a factorisation by expanding it again, or use the Factoring Calculator. For two brackets multiplied together, such as (x + 2)(x + 5), see expanding double brackets.

Common mistakes

• Only multiplying the first term. 5(x + 3) is 5x + 15, not 5x + 3.
• Losing a negative. −4(x − 5) gives +20, not −20.
• Subtracting a bracket. In −2(x − 1), both terms change sign: −2x + 2.
• Adding instead of multiplying powers. x × x is x², not 2x.
• Collecting unlike terms. x² and x are not like terms, so x² + 9x cannot be simplified further.

Practise with a short timed paper

Expanding brackets comes up throughout GCSE Maths. The daily GCSE Maths Foundation and GCSE Maths Higher papers are short timed papers of 8 questions, with a new one every day.

Read next

Maths · AlgebraExpanding double brackets: FOIL, grid method and examplesHow to expand double brackets with FOIL and the grid method, from (x + 2)(x + 6) to squared brackets, difference of two squares and triple brackets.