Learn how to expand single and double brackets, factorise expressions, and complete the square. Complete GCSE algebra guide with worked examples for all exam boards.
Expanding a single bracket means multiplying the term outside the bracket by every term inside. This is sometimes called the distributive law.
Expand 3(2x + 5)
To expand two brackets multiplied together, use FOIL: First, Outer, Inner, Last. Multiply each term in the first bracket by each term in the second, then collect like terms.
Expand (x + 3)(x − 2)
Factorising is the reverse of expanding. Find the highest common factor (HCF) of all terms, then write it outside a bracket. Each term inside the bracket is what remains after dividing by the HCF.
Factorise 12x² + 8x
For quadratics in the form x² + bx + c, find two numbers that multiply to give c and add to give b. These go into the two brackets.
Factorise x² + 7x + 12
A special factorisation pattern: a² − b² = (a + b)(a − b). Spotting this pattern saves time in exams.
Expanding removes brackets (e.g., 3(x + 2) → 3x + 6). Factorising introduces brackets (e.g., 3x + 6 → 3(x + 2)). They're inverse operations.
Use the "ac method": for ax² + bx + c, find two numbers that multiply to a × c and add to b. Then factorise by grouping. This is Higher tier content.
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. It's used to find the vertex of a parabola and to solve quadratics when factorising is difficult.
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