GCSE Maths: Expanding and Factorising

Expanding and factorising are the workhorse skills of GCSE algebra: almost every equation-solving, graph-sketching and proof question eventually leans on them. This quiz builds the full toolkit in the order the skills naturally stack.

It starts with collecting like terms and expanding single brackets — the places where sign errors are born — then moves to expanding double brackets with the full four-term working, so that the middle terms are added consciously rather than by half-remembered pattern. From there the direction reverses: factorising into single brackets by extracting the highest common factor, then factorising quadratics of the form x² + bx + c by finding the pair of numbers with the right product and sum, including negative cases like x² − 7x + 10 where both signs must be handled correctly.

The difference of two squares gets deliberate attention from both directions: recognising which expressions fit the a² − b² pattern (including 4x² − 25, where the coefficient must itself be a perfect square) and expanding conjugate pairs like (x − 5)(x + 5) to see why the middle terms vanish. A combined-skills question — expanding and simplifying 5(x + 2) − 3(x − 1) — checks the most common exam slip of all: mishandling the negative sign distributed across the second bracket.

Every question includes a worked explanation, and the distractors reflect real student errors rather than random values: dropped middle terms, wrong factor signs, and half-distributed brackets. This set is the natural warm-up for the Linear and Quadratic Equations quiz in the same collection, where these manipulations get put to work solving actual equations.

• Collect like terms and expand single brackets without sign errors
• Expand double brackets systematically, keeping all four terms before simplifying
• Factorise expressions into single brackets using the highest common factor
• Factorise quadratics of the form x² + bx + c, handling negative coefficients correctly
• Recognise and use the difference of two squares in both directions

Topic scope follows the Algebra strand of the Department for Education's GCSE mathematics subject content: simplifying and manipulating algebraic expressions by collecting like terms, expanding products of one or two binomials, and factorising quadratic expressions including the difference of two squares.