Factorising is the reverse of multiplying polynomials. Expanding turns (x + 3)(x + 4) into x^2 + 7x + 12; factorising goes the other way — it takes a quadratic and writes it as a product of two brackets.

This lesson covers the simplest case: a quadratic x^2 + bx + c whose leading coefficient is 1. (If every term shares a number first, pull out the common factor before you start.)

The two-number trick

Look at what happens when we expand (x + p)(x + q) and collect like terms:

So the number in front of x is p + q, and the constant on the end is pq. To factorise x^2 + bx + c we simply run that backwards — find two numbers that multiply to c and add to b:

Worked example

To factorise x^2 + 7x + 12 we need two numbers that multiply to 12 and add to 7. List the factor pairs of 12 and check each sum:

The pair 3 and 4 works, so p = 3 and q = 4:

Multiply the brackets back out to check — you should land on x^2 + 7x + 12 again.

See it as an area model

The four products fit together as the area of a rectangle. The x^2 is a square of side x; the 7x is two strips (3x and 4x); and the 12 is a 3 \times 4 block. Slide them together and the whole rectangle is (x + 3) wide and (x + 4) tall — which is exactly the factorisation. Step through it.