Algebraic manipulation

• Use letters for generalised numbers and substitute values into expressions and formulas.
• Simplify by collecting like terms, e.g. 2a² + 3ab − 1 + 5a² − 9ab + 4 = 7a² − 6ab + 3.
• Expand products of brackets, including three brackets.
• Factorise fully: common factors, grouping, the difference of two squares, perfect squares, quadratics and ax³ + bx² + cx.
• Complete the square for ax² + bx + c.

Replace each letter with its value, in brackets

In algebra a letter represents a number. 3n means 3 × n, n² means n × n and ab means a × b. To substitute, replace each letter with its value in brackets (this keeps negative signs safe), then follow the order of operations.

Same letters, same powers

Like terms have exactly the same letters with the same powers: 2a² and 5a² are like terms; a² and a are not. Add or subtract their coefficients, keeping the sign in front of each term with it.

Every term in one bracket times every term in the other

Multiply each term inside by the term outside. For two brackets, multiply every term in the first by every term in the second, then collect like terms. For three brackets, expand two of them first, then multiply the result by the third.

Factorise means factorise fully

Take out the HCF of the numbers and of the letters. Check by expanding. For four terms of the form ax + bx + kay + kby, group them in pairs, factorise each pair, then take out the common bracket.

Recognise the pattern first

For ax² + bx + c with a ≠ 1: find two numbers that multiply to ac and add to b, split the middle term, then group.