Expanding brackets: single, double and triple brackets
Expanding brackets means multiplying out the brackets so an expression has none left. To expand a single bracket, multiply every term inside by the term outside: 3(x + 4) = 3x + 12. To expand double brackets, multiply every term in the first bracket by every term in the second, then collect like terms. Triple brackets are done two at a time.
How to expand brackets
To expand brackets, multiply each term inside the bracket by whatever is outside it. The number or letter in front of a bracket multiplies everything inside, not just the first term. Then, if there's more than one bracket, collect the like terms to simplify the answer.
The idea isn't new. In Year 4, the national curriculum has children write statements like 39 × 7 = 30 × 7 + 9 × 7: splitting a number into parts and multiplying each part. That's partitioning, and it's exactly what expanding a bracket does. 3 × 14 is 3 × 10 and 3 × 4, which is 30 and 12, so 3 × 14 = 42. In algebra, 3(x + 4) is 3 × x and 3 × 4, so it's 3x + 12.
Schools teach brackets as part of KS3 maths, across Years 7 to 9 in the order each school chooses, and then use them all the way through GCSE.
Expanding single brackets
To expand a single bracket, multiply the term outside by each term inside, keeping the signs.
Worked example: a single bracket
Expand 5(2y − 3).
- 5 × 2y is 10y.
- 5 × −3 is −15.
So 5(2y − 3) = 10y − 15.
The term outside can be a letter, or a negative number:
- x(x + 6) = x2 + 6x, because x × x is x2.
- −2(x − 4) = −2x + 8, because −2 × −4 is positive 8. Two signs the same make a positive, as the negative numbers guide explains.
Expanding and simplifying brackets
Expanding and simplifying means multiplying out every bracket, then collecting like terms. The algebra basics guide covers collecting like terms.
Worked example: expand and simplify
Expand and simplify 2(x + 5) + 3(x − 1).
- Expand the first bracket: 2x + 10.
- Expand the second bracket: 3x − 3.
- Put them together: 2x + 10 + 3x − 3.
- Collect like terms: 2x and 3x make 5x, and 10 take away 3 is 7.
The answer is 5x + 7. A quick check: put x = 1 into both. The question gives 2 lots of 6 and 3 lots of 0, which is 12. The answer gives 5 and 7, which is also 12.
How to expand double brackets
To expand double brackets (two brackets multiplied together), multiply each term in the first bracket by each term in the second. With two terms in each bracket, that's four multiplications. Then collect the like terms. A grid keeps track of them, the same way the grid method does for multiplying numbers.
Worked example: expanding two brackets with a grid
Expand (x + 3)(x + 5). Put the terms of one bracket across the top and the other down the side, then multiply to fill each box:
| × | x | +3 |
|---|---|---|
| x | x2 | 3x |
| +5 | 5x | +15 |
Add up the four boxes: x2 + 3x + 5x + 15. The 3x and 5x are like terms, so the answer is x2 + 8x + 15.
Worked example: double brackets with a minus sign
Expand and simplify (x + 4)(x − 2).
| × | x | −2 |
|---|---|---|
| x | x2 | −2x |
| +4 | 4x | −8 |
The four boxes give x2 − 2x + 4x − 8. Collect the x terms: −2x and 4x make 2x. The answer is x2 + 2x − 8.
Some children learn to do the four multiplications without a grid: first terms, outside terms, inside terms, last terms. It's the same four multiplications either way.
Squaring a bracket
(x + 3)2 means (x + 3)(x + 3), so it's a double bracket. The four boxes are x2, 3x, 3x and 9, which gives x2 + 6x + 9. It is not x2 + 9: that misses out the two middle boxes.
How to expand triple brackets
To expand triple brackets, expand two of the brackets first, then multiply the answer by the third bracket. The key stage 3 curriculum includes expanding "two or more" brackets. At GCSE, the DfE content marks the "or more" part as content that only the higher tier is assessed on, so expanding three brackets is a higher tier skill, while double brackets are for everyone.
Worked example: expanding three brackets
Expand (x + 1)(x + 2)(x + 3).
- Expand the first two: (x + 1)(x + 2) gives x2 + 2x + x + 2, which is x2 + 3x + 2.
- Multiply that by (x + 3). The grid now has three rows:
| × | x | +3 |
|---|---|---|
| x2 | x3 | 3x2 |
| +3x | 3x2 | 9x |
| +2 | 2x | +6 |
- Add the six boxes and collect like terms: x3 + 6x2 + 11x + 6.
Check with x = 1: the question is 2 × 3 × 4, which is 24. The answer is 1 + 6 + 11 + 6, which is also 24.
Expanding brackets runs backwards too: putting brackets back in is factorising, and children use both when they solve equations with brackets.
Common mistakes when expanding brackets
- Only multiplying the first term. 4(x + 3) is 4x + 12, not 4x + 3.
- Dropping a sign. In 5(2y − 3), the 3 is negative, so 5 lots of it is −15.
- Getting the sign wrong with a negative outside. −3(x − 2) is −3x + 6, because a negative times a negative is positive.
- Squaring a bracket as two squares. (x + 4)2 is x2 + 8x + 16, not x2 + 16.
- Missing a box. Double brackets with two terms each always give four terms before simplifying. Count them.
- Adding unlike terms. x2 and x can't be collected together.
How to help at home
- Start from numbers. Ask your child to work out 6 × 23 as 6 × 20 and 6 × 3. Then show them 6(x + 3) works the same way.
- Ask for the grid. If double brackets go wrong, drawing the grid usually shows where.
- Check with a number. Put x = 1 or x = 2 into the question and the answer. If they don't match, something has gone wrong.
- Keep it short. Two or three questions done carefully beat a page of rushed ones.
Expanding brackets questions to try
- Expand 4(x + 5).
- Expand 3(2a − 7).
- Expand and simplify 2(x + 3) + 4(x + 1).
- Expand and simplify (x + 2)(x + 7).
- Expand and simplify (x − 5)(x + 3).
- Higher tier challenge: expand (x + 1)(x + 2)(x + 4).
Answers
- 4x + 20
- 6a − 21
- 2x + 6 + 4x + 4, which simplifies to 6x + 10
- x2 + 7x + 2x + 14, which simplifies to x2 + 9x + 14
- x2 + 3x − 5x − 15, which simplifies to x2 − 2x − 15
- (x + 1)(x + 2) is x2 + 3x + 2. Multiplying by (x + 4) gives x3 + 4x2 + 3x2 + 12x + 2x + 8, which simplifies to x3 + 7x2 + 14x + 8.
For the rest of what children learn in Years 7 to 9, see the KS3 maths guide.
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Sources
We check every method and fact on this page against these official documents.
- National curriculum in England: mathematics programmes of study (Department for Education). Used for: key stage 3: multiplying a single term over a bracket, expanding products of two or more binomials and collecting like terms; Year 4 notes and guidance: the distributive law (39 × 7 = 30 × 7 + 9 × 7); key stage 3 covers Years 7 to 9 without setting an order.
- GCSE mathematics: subject content and assessment objectives (Department for Education). Used for: expanding products of two binomials is standard content for every student; the words "or more" (three or more brackets) are in bold, which the document says only the more highly attaining students are assessed on, the higher tier.