12.2 Factorising harder quadratics — Learn

Prerequisites: expanding brackets and factorising monic quadratics.

Objectives: factorise $ax^2+bx+c$ when $a$ is not $1$, including cases where the leading coefficient is negative. This extends A4.

Split the middle term

For $ax^2+bx+c$, find two numbers with product $ac$ and sum $b$. Split $bx$, then factor by grouping.

Example 1. Factorise $6x^2+11x+3$.

The product $ac$ is $18$. The numbers $9$ and $2$ have product $18$ and sum $11$.

Negative leading coefficient

Take out a negative factor first. Then use the same method.

Example 2. Factorise $-2x^2+5x+3$.

$$ -2x^2+5x+3=-(2x^2-5x-3) $$ The product is $-6$ and the sum is $-5$, so use $-6$ and $1$.

Watch out. Choosing numbers with the right product is not enough. Their sum must match the middle coefficient too. Expand your final factors to check both.

You should now be able to

• use the product $ac$ and sum $b$ to split a middle term;
• factorise by grouping matching brackets;
• handle a negative leading coefficient by taking out $-1$ first.