Factorising Quadratics
Grade 5What you are looking for
To factorise x² + bx + c, you need two numbers that add to b and multiply to c. Call them p and q. Then the factorised form is (x + p)(x + q).
This works because expanding (x + p)(x + q) gives x² + (p + q)x + pq. So the sum and product of your two numbers must match b and c.
Worked example
When the coefficient of x² is not 1
For ax² + bx + c where a ≠ 1, you need to find two numbers that multiply to a × c and add to b. Then split the middle term and factorise by grouping. This is called the ac method.
Always check
Try this one
(a) Factorise \( x^2 - 9x + 18 \). [2]
(b) Hence solve \( x^2 - 9x + 18 = 0 \). [1]
Part (a) uses factorising a quadratic with a positive constant and a negative middle coefficient.
Ask yourself: 'Which two negative numbers multiply to give \( +18 \) and add to give \( -9 \)?'
Once you have factorised in part (a), use the fact that if a product of two brackets equals zero, then at least one bracket must equal zero.
