Quadratic Functions and Equations in One Variable
How to Factorise a quadratic
Use factorising to solve a quadratic quickly when it splits into neat brackets.
Before you start
- Writing a quadratic in the form ax² + bx + c = 0
- Finding factor pairs of a number
- Working with positive and negative numbers
When to use it
Use factorising to solve a quadratic quickly when it splits into neat brackets.
The steps
- Write the equation as ax² + bx + c = 0.
- Find two numbers that multiply to a×c and add to b.
- Split the middle term using those two numbers.
- Factorise in pairs to get two brackets.
- Set each bracket equal to zero to find the two roots.
Worked example
Solve x² + 7x + 12 = 0 by factorising.
- The equation is already in the form ax² + bx + c = 0 with a = 1, b = 7, c = 12.
- Find two numbers that multiply to a × c = 12 and add to b = 7: these are 3 and 4.
- Split the middle term: x² + 3x + 4x + 12 = 0.
- Factorise in pairs: x(x + 3) + 4(x + 3) = 0, giving (x + 3)(x + 4) = 0.
- Set each bracket to zero: x + 3 = 0 gives x = −3, and x + 4 = 0 gives x = −4.
A second example, with a twist
The coefficient of x² is not 1 and c is negative, so one of the roots turns out to be a fraction. Solve 3x² − 5x − 2 = 0 by factorising.
- The equation is in the form ax² + bx + c = 0 with a = 3, b = −5, c = −2.
- Find two numbers that multiply to a × c = 3 × (−2) = −6 and add to b = −5: these are −6 and 1.
- Split the middle term: 3x² − 6x + x − 2 = 0.
- Factorise in pairs: 3x(x − 2) + 1(x − 2) = 0, giving (x − 2)(3x + 1) = 0.
- Set each bracket to zero: x − 2 = 0 gives x = 2, and 3x + 1 = 0 gives x = −1/3.
Practise this in a KBAT problem
Frequently asked questions
What if I cannot find two whole numbers that work?
Then the quadratic does not factorise neatly, and you should switch to the quadratic formula or completing the square instead. Factorising only works when nice whole numbers multiply to a × c and add to b.
If you have tried the factor pairs and none fit, stop and use another method.
Why do I set each bracket equal to zero?
If two things multiply to give zero, at least one of them must be zero. Once the quadratic is written as (bracket)(bracket) = 0, each bracket is a factor that could be the zero one.
Setting each to zero gives the two values of x that make the whole product zero.
Does the order of the two numbers matter when splitting?
No. When you split the middle term, writing the two numbers in either order gives the same final brackets.
For 3x² − 5x − 2 you can write −6x + x or x − 6x; both factorise to the same answer. Just be careful to keep each number's sign attached to it.