Expanding brackets and factorising · GCSE Maths

Expanding single brackets

Multiply every term in the first bracket by every term in the second (FOIL or a grid): (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.
Squared brackets: (x − 4)² = (x − 4)(x − 4) = x² − 8x + 16, not x² + 16.
(x + 5)(x − 5) = x² − 25: the middle terms cancel.

Factorising is the reverse of expanding. Take out the highest common factor: 6x + 9 = 3(2x + 3); 4x² + 8x = 4x(x + 2).
'Factorise fully' means take out every common factor.

For x² + bx + c, find two numbers that multiply to c and add to b: x² + 7x + 12 = (x + 3)(x + 4).
Signs: x² − x − 6 = (x − 3)(x + 2), because −3 × 2 = −6 and −3 + 2 = −1.
Difference of two squares: x² − 9 = (x + 3)(x − 3).
(Higher) For ax² + bx + c, find two numbers that multiply to ac and add to b, then split the middle term: 2x² + 7x + 3 = (2x + 1)(x + 3).

Simplify by factorising the top and the bottom, then cancelling common factors: (x² − 9)/(x + 3) = (x + 3)(x − 3)/(x + 3) = x − 3.
Add or subtract using a common denominator: 1/x + 2/(x + 1) = (x + 1 + 2x)/(x(x + 1)) = (3x + 1)/(x(x + 1)).
Multiply and divide like number fractions: to divide, multiply by the reciprocal of the second fraction. Factorise first so you can cancel.
To solve an equation with algebraic fractions, multiply every term by the common denominator: 3/(x − 1) = 2 → 3 = 2(x − 1) → x = 2.5.

Expand: Multiply out brackets.
Factorise: Put an expression into brackets by taking out common factors.
Highest common factor: The largest factor shared by all the terms.
Quadratic expression: An expression where the highest power of x is 2.
Difference of two squares: An expression of the form a² − b², which factorises to (a + b)(a − b).
Algebraic fraction: A fraction with algebraic expressions on the top, the bottom or both.