How to Factorise Harder Quadratics

📋 Spec coverage: GCSE AQA Mathematics (8300) This article covers:

• AQA A4 (Higher): factorising quadratic expressions of the form $a{x}^2+bx+c$.

Tier: Higher

To factorise a harder quadratic, multiply the coefficient of $x^2$ by the constant term. Find two numbers that multiply to this result and add to the coefficient of $x$. Then split the middle term and factorise by grouping.

The method

For a quadratic in the form

$$a{x}^2+bx+c,$$

follow these steps:

1. Calculate $a\times c$.
2. Find two numbers that multiply to $ac$ and add to $b$.
3. Split $bx$ using those two numbers.
4. Factorise the first two terms and the last two terms separately.
5. Take out the common bracket.

To factorise means to rewrite an expression as a product of brackets.

The coefficient is the number multiplying a variable, so in $6x^2+11x+3$, $a=6$, $b=11$ and $c=3$. “Grouping” means arranging terms in pairs so that each pair contains the same bracket.

Worked example

Factorise:

$$6x^2+11x+3.$$

First, multiply the coefficient of $x^2$ by the constant:

$$6\times 3=18.$$

We need two numbers that multiply to $18$ and add to $11$. These are $9$ and $2$.

Split the middle term:

$$6x^2+9x+2x+3.$$

Factorise each pair:

$$3x(2x+3)+1(2x+3).$$

Both parts contain $(2x+3)$, so:

$$6x^2+11x+3=(3x+1)(2x+3).$$

Check by expanding the brackets: the middle terms are $9x$ and $2x$, which add to $11x$.

Why this works

Splitting $11x$ into $9x+2x$ does not change the expression, because $9x+2x=11x$. It reveals a common factor in each pair. Once the identical bracket appears twice, the distributive law lets you take it outside.

A quick multiplication check confirms all three terms, not only the first and last terms.

Common exam mistake

Do not just look for two numbers that multiply to $c$. When $a\ne 1$, they must multiply to $a\times c$. Also check the signs carefully when $b$ or $c$ is negative.

Completeness check: The Higher-tier skill of factorising quadratics in the form $a{x}^2+bx+c$ is covered with a complete method, example and expansion check. Other AQA A4 skills are not claimed here.