Quadratics — factorising and roots
Maths · GCSE · The School
The factor-pair hunt
To factorise x² + 7x + 12, find two numbers that MULTIPLY to 12 and ADD to 7: that is 3 and 4. So x² + 7x + 12 = (x + 3)(x + 4). Signs follow the c-term: c positive means same signs (both matching b); c negative means opposite signs.
From factors to solutions
If (x + 3)(x + 4) = 0, then one bracket must be zero — a product is zero only when a factor is. So x = −3 or x = −4. This "zero-product" step is why we factorise at all.
Roots are crossing points
Graph y = x² + 7x + 12 and it is a parabola crossing the x-axis exactly at x = −3 and x = −4 — the roots. The graph and the algebra are the same fact in two languages: solving the equation finds the crossings; reading the crossings solves the equation.
Two cases the factor-pair hunt does not cover
First, the difference of two squares. x² − 9 looks unfactorisable because there is no x term, but it is (x + 3)(x − 3): the middle terms cancel when you expand. Any a² − b² factorises this way, and it is worth recognising on sight because it appears constantly. Second, rearranging. x² + 5x = 14 is not ready — the zero-product rule works only at zero — so subtract 14 from both sides to get x² + 5x − 14 = 0, then hunt the pair that multiplies to −14 and adds to 5: that is 7 and −2, giving (x + 7)(x − 2) = 0 and x = −7 or x = 2. Check by substituting the roots back into the ORIGINAL equation: 4 + 10 = 14, so x = 2 works.