To expand a single bracket, multiply the term outside by every term inside: 3(x + 4) = 3x + 12. With more than one bracket, expand each and then collect like terms, taking care with negative signs. To expand two brackets, multiply each term in the first by each term in the second, giving four products, then simplify.
Expanding brackets
- a(b + c) = ab + ac
- −a(b − c) = −ab + ac
- (a + b)(c + d) = ac + ad + bc + bd
- (x + a)² = x² + 2ax + a²
Expanding a single bracket
Multiply the term outside the bracket by every term inside it. 3(x + 4) = 3x + 12. An area model helps: draw a rectangle with width 3, split into lengths x and 4, so the two parts have areas 3x and 12. The term outside can be a letter too: x(x + 6) = x² + 6x.
Expanding and simplifying
When there are several brackets, expand each one, then collect like terms. Take particular care with a negative number in front of a bracket: it multiplies every term inside. −2(x − 4) = −2x + 8, because −2 × −4 = +8.
Expanding double brackets
Multiply each term in the first bracket by each term in the second, which gives four products. A grid or the word FOIL (first, outside, inside, last) keeps track of them. (x + 4)(x + 6) = x² + 6x + 4x + 24 = x² + 10x + 24. The two middle terms usually combine.
Negative terms and coefficients
Keep the sign with each term when multiplying. (x − 5)(x + 2) gives x², +2x, −5x and −10, which collect to x² − 3x − 10. If a bracket has a coefficient of x, such as (2x + 1)(x − 3), the first product is 2x².
Squaring a bracket
(x + 3)² means (x + 3)(x + 3), not x² + 9. Expanding gives x² + 3x + 3x + 9 = x² + 6x + 9. The middle term, 2 × 3 × x, is the one most often missed.
Worked examples
Example 1. Expand 3(x + 4).
- 3 × x = 3x.
- 3 × 4 = 12.
Answer: 3x + 12.
Example 2. Expand and simplify 2(x + 3) + 3(x + 1).
- Expand: 2x + 6 + 3x + 3.
- Collect like terms.
Answer: 5x + 9.
Example 3. Expand and simplify 5(2x − 1) − 2(x − 4).
- 5(2x − 1) = 10x − 5.
- −2(x − 4) = −2x + 8.
- Collect: 10x − 2x − 5 + 8.
Answer: 8x + 3.
Example 4. Expand (x + 4)(x + 6).
- x × x = x², x × 6 = 6x, 4 × x = 4x, 4 × 6 = 24.
- Collect the middle terms.
Answer: x² + 10x + 24.
Example 5. Expand (2x + 1)(x − 3).
- 2x × x = 2x², 2x × (−3) = −6x, 1 × x = x, 1 × (−3) = −3.
- Collect: −6x + x = −5x.
Answer: 2x² − 5x − 3.
Example 6. Expand (x + 3)².
- (x + 3)(x + 3) = x² + 3x + 3x + 9.
Answer: x² + 6x + 9.
Common mistakes
- Multiplying only the first term. 3(x + 4) = 3x + 12, not 3x + 4.
- Sign errors with a negative outside. −2(x − 4) = −2x + 8.
- Missing products in double brackets. There are four products; check you have them all.
- Writing (x + 3)² = x² + 9. Expand fully to get x² + 6x + 9.
- Not collecting like terms. Combine the two middle terms at the end.
How expanding brackets is examined
Expanding brackets is tested on every Edexcel GCSE (1MA1) and International GCSE (4MA1) paper in some form; single brackets are Foundation, and double brackets appear at both tiers. Questions ask you to:
- expand a single bracket, including with a letter outside
- expand and simplify two or more brackets
- expand double brackets, including squared brackets
- expand three brackets (Higher tier)
Use the simplifying and expanding worksheet with answers, and generate more questions in the algebra revision engine.
Expanding is the reverse of factorising, and it is used in quadratic equations and algebraic proof.
Frequently asked questions
How do you expand brackets?
Multiply every term inside the bracket by the term outside.
How do you expand double brackets?
Multiply each term in the first bracket by each term in the second, then collect like terms.
What is (x + 5)² expanded?
x² + 10x + 25.