Expanding & Factorising Brackets (Cambridge IGCSE Maths): Revision notes

Section 1

How do you expand single and double brackets?

Expanding a single bracket means multiplying every term inside by the term outside.

$3x(2x - 4y) = 6x^2 - 12xy$

Expanding double brackets means multiplying every term in the first bracket by every term in the second (often remembered as FOIL: First, Outer, Inner, Last).

$(2x+1)(x-4) = 2x^2 - 8x + x - 4 = 2x^2 - 7x - 4$

Key termsexpand

Common mistake

Forgetting to multiply the second term of the second bracket is the most common error — always produce all four products before simplifying.

Section 2

How do you expand three or more brackets?

Expand two brackets first, then multiply the result by the remaining bracket.

$(x-2)(x+3)(2x+1)$

Expand the first pair: $(x-2)(x+3) = x^2 + x - 6$
Multiply by the third bracket: $(x^2+x-6)(2x+1)$
Distribute and collect like terms: $2x^3 + x^2 + 2x^2 + x - 12x - 6 = 2x^3 + 3x^2 - 11x - 6$

Exam tip

Work systematically: expand only two brackets at a time, fully simplifying before bringing in the next bracket.

Section 3

How do you factorise using common factors and grouping?

Factorising is the reverse of expanding: writing an expression as a product of factors.

Extract the highest common factor (HCF) fully: $9x^2 + 15xy = 3x(3x + 5y)$
Factorise by grouping when there are four terms: $ax + bx + kay + kby = x(a+b) + ky(a+b) = (a+b)(x+ky)$

Key termsfactorisehighest common factor

Exam tip

Always check factorisation by expanding your answer back out — it should match the original expression exactly.

Section 4

How do you factorise special quadratic forms?

Form	Example
Difference of two squares: $a^2x^2 - b^2y^2 = (ax-by)(ax+by)$	$x^2 - 9 = (x-3)(x+3)$
Perfect square: $a^2 + 2ab + b^2 = (a+b)^2$	$x^2 + 6x + 9 = (x+3)^2$
$ax^2 + bx + c$	$x^2 + 5x + 6 = (x+2)(x+3)$
$ax^3 + bx^2 + cx$	$2x^3 + 6x^2 = 2x^2(x+3)$

For $ax^2+bx+c$ , find two numbers that multiply to give $ac$ and add to give $b$ .

Key termsdifference of two squaresperfect square

Example

Factorise $x^2 - 5x - 14$ : find two numbers multiplying to $-14$ and adding to $-5$ : $-7$ and $2$ , giving $(x-7)(x+2)$ .

Section 5

How do you complete the square?

To write $ax^2 + bx + c$ in the form $a(x+p)^2 + q$ :

Factor out $a$ if $a \neq 1$
Halve the coefficient of $x$ to get $p$
Write $(x+p)^2$ , then subtract $p^2$ and add the original constant $c$

For example, $x^2 + 6x + 5 = (x+3)^2 - 9 + 5 = (x+3)^2 - 4$ .

Key termscomplete the square

Exam tip

Completing the square is Extended-tier content and is essential for finding a quadratic's turning point and for one method of solving quadratic equations.

Must Know

Expand by multiplying every term inside a bracket by the term(s) outside
Always extract the highest common factor fully first when factorising
Difference of two squares: $a^2-b^2=(a-b)(a+b)$ ; perfect square: $a^2+2ab+b^2=(a+b)^2$
For $ax^2+bx+c$ , find two numbers multiplying to $ac$ and summing to $b$
Completing the square: $x^2+bx+c = (x+\frac{b}{2})^2 - (\frac{b}{2})^2 + c$
Always verify a factorisation by expanding it back out

That's the notes covered.

Carry on to the next subtopic.