What you'll learn
Expanding brackets means multiplying out terms to remove brackets, a key algebraic skill used throughout GCSE Maths. In this guide you will learn how to expand single brackets, how to expand and simplify double brackets, how to square a bracket, how to expand three brackets, and how to avoid sign errors. These skills are essential for solving equations, factorising and rearranging formulae.
Key terms and definitions
Expand — to multiply out brackets and write the expression without them.
Term — a part of an expression separated by + or − signs.
Coefficient — the number multiplying a variable.
Like terms — terms with the same variable part that can be added or subtracted.
Binomial — an expression with two terms, such as (x + 3).
Core concepts
Expanding a single bracket
To expand a single bracket, multiply every term inside by the term outside. For example, 3(x + 4) = 3 × x + 3 × 4 = 3x + 12. Take care with signs: −2(x − 5) = −2x + 10 (a negative times a negative gives a positive).
Expanding double brackets
To expand two brackets, multiply each term in the first bracket by each term in the second (often remembered as FOIL: First, Outer, Inner, Last), then collect like terms. For example:
(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.
Always combine the two middle terms.
Squaring a bracket
A squared bracket means the bracket multiplied by itself: (x + 4)² = (x + 4)(x + 4). Expand as usual: x² + 4x + 4x + 16 = x² + 8x + 16. A common error is writing (x + 4)² = x² + 16 — you must include the middle term 2 × x × 4 = 8x.
Negative signs in double brackets
Watch signs carefully. (x − 2)(x − 6) = x² − 6x − 2x + 12 = x² − 8x + 12. Two negatives multiply to a positive, so the last term is +12. Keep each multiplication's sign correct before collecting.
Expanding three brackets
To expand three brackets, expand two of them first, then multiply the result by the third bracket, multiplying every term. Simplify at the end by collecting like terms. Work in stages to avoid mistakes.
Worked examples
Example 1: Single bracket
Expand 5(2x − 3).
5 × 2x = 10x and 5 × (−3) = −15, so the answer is 10x − 15.
Example 2: Double bracket
Expand and simplify (x + 7)(x − 2).
x² − 2x + 7x − 14 = x² + 5x − 14.
Example 3: Squaring
Expand (2x + 3)².
(2x + 3)(2x + 3) = 4x² + 6x + 6x + 9 = 4x² + 12x + 9.
Common mistakes and how to avoid them
Forgetting to multiply every term. The outside term must multiply each term inside the bracket.
Sign errors. Track the sign of each term; negative × negative = positive.
Squaring incorrectly. (a + b)² is not a² + b²; include the middle term 2ab.
Not collecting like terms. After expanding double brackets, combine the two middle terms.
Rushing three brackets. Expand two first, then multiply by the third in stages.
Exam technique for Expanding Brackets
Multiply systematically — every inside term by the outside term (or FOIL for doubles).
Track signs carefully throughout.
Always simplify by collecting like terms at the end.
Treat squares as two brackets, never forgetting the middle term.
Break three brackets into stages to stay accurate.
Quick revision summary
Expanding brackets removes them by multiplication. For a single bracket, multiply every term inside by the term outside (3(x + 4) = 3x + 12), watching signs. For double brackets, multiply each term in the first by each term in the second (FOIL: First, Outer, Inner, Last), then collect like terms ((x + 3)(x + 5) = x² + 8x + 15). Squaring a bracket means multiplying it by itself, so (x + 4)² = x² + 8x + 16 — never forget the middle term 2ab. Be careful with negative signs: negative × negative = positive. For three brackets, expand two first, then multiply the result by the third, simplifying at the end. The biggest pitfalls are missing a multiplication, sign slips, and the (a + b)² error. Multiply every term, track signs, collect like terms, and work in stages for harder cases — these habits make expanding reliable and set you up for factorising and solving equations.