Factorising Trinomials

A trinomial like $x^2 + bx + c$ factorises into two brackets $(x + p)(x + q)$, where you must find two numbers $p$ and $q$ that

• multiply to give $c$ (the constant), and
• add to give $b$ (the coefficient of $x$).

Worked example 1. Factorise $x^2 + 5x + 6$. Find two numbers that multiply to $6$ and add to $5$: they are $2$ and $3$. So $x^2 + 5x + 6 = (x + 2)(x + 3)$.

Worked example 2 (with negatives). Factorise $x^2 - x - 12$. Two numbers that multiply to $-12$ and add to $-1$: they are $-4$ and $3$ (since $-4 \times 3 = -12$ and $-4 + 3 = -1$). So $x^2 - x - 12 = (x - 4)(x + 3)$.

Check by expanding: $(x-4)(x+3) = x^2 + 3x - 4x - 12 = x^2 - x - 12$ ✓.