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.
Completing the square gives the turning point
Written in the form of a bracket squared plus a number, the turning point is read straight off with the sign inside the bracket reversed. It also solves equations that will not factorise, and it is where the quadratic formula comes from. Worth the practice for the turning point alone.
The discriminant counts the roots
The part under the root sign decides everything: positive gives two distinct roots, zero gives one repeated root, negative gives none in the real numbers. Questions asking how many times a curve crosses the axis are asking for this, and the sign alone is the answer.