Coaching Centre
Factorising quadratic trinomials
Factorising monic trinomials x² + bx + c using two numbers that multiply to c and add to b, and non-monic trinomials ax² + bx + c by splitting the middle term.
Monic trinomials
Factorise $x^2 - 2x - 15$.
Multiply to $-15$, add to $-2$: $-5$ and $3$.
$(x - 5)(x + 3)$
Non-monic trinomials
Factorise $3x^2 - 10x - 8$.
$a \times c = -24$. Numbers that multiply to $-24$ and add to $-10$: $-12$ and $2$.
$3x^2 - 12x + 2x - 8 = 3x(x - 4) + 2(x - 4)$
$= (3x + 2)(x - 4)$
$x^2 + 7x + 12$
$(x + 3)(x + 4)$
$x^2 + x - 6$
$(x + 3)(x - 2)$
Non-monic first step
Find $a \times c$.
Factorise $x^2 - 2x - 15$.
$-5 \times 3 = -15$, $-5 + 3 = -2$.
Factorise $3x^2 - 10x - 8$.
Split as $-12x + 2x$.
Factorise $x^2 + 7x + 12$.
$3 \times 4 = 12$, $3 + 4 = 7$.
Factorise $x^2 - 9x + 20$.
$-4$ and $-5$.
Factorise $x^2 + x - 6$.
$3$ and $-2$.
Factorise $2x^2 + 10x + 12$ fully.
$2(x^2 + 5x + 6)$.
Factorise $2x^2 + 7x + 3$.
$ac = 6$: $6$ and $1$.
Factorise $6x^2 + x - 2$.
$ac = -12$: $4$ and $-3$.
New numbers every time. Keep going until you get five in a row.