KS4 Maths · Algebra · Year 10
Factorising quadratics
Turn x² + bx + c into two brackets — and use it to solve equations.
About 25 minutes
Step 1 · Wonder why it matters
Hook
The reverse of expanding
You already know how to expand (x + 2)(x + 3) to get x² + 5x + 6. But what if a question gives you x² + 5x + 6 and asks for the brackets?
Factorising quadratics is one of the most common GCSE algebra skills, and it unlocks solving quadratic equations.
Step 2 · See it
Picture
See the area model
Expanding (x + 2)(x + 3) is like finding the area of a rectangle with sides x + 2 and x + 3. The four parts add up to the quadratic.
| × | x | +3 |
|---|---|---|
| x | x² | 3x |
| +2 | 2x | 6 |
Step 3 · Understand how it works
Logic
Find two numbers
For x² + bx + c, find two numbers that multiply to give c and add to give b.
Those two numbers go in the brackets: (x + p)(x + q).
Watch the signs. If c is positive, both numbers have the same sign as b. If c is negative, the numbers have opposite signs.
To solve x² + bx + c = 0, factorise and then set each bracket equal to zero.
- Quadratic
- An expression where the highest power of x is x².
- Factorise
- Write as a product of factors — put into brackets.
- Root
- A value of x that makes the quadratic equal zero.
Worked example
Solve x² − x − 12 = 0
- 1Find two numbers that multiply to −12 and add to −1.
- 2Try factor pairs of 12: 3 and 4 differ by 1. We need −4 and +3, since −4 × 3 = −12 and −4 + 3 = −1.
- 3So x² − x − 12 = (x − 4)(x + 3).
- 4Set each bracket to zero: x − 4 = 0 or x + 3 = 0.
Answer: x = 4 or x = −3
Step 4 · Have a go
Do
Have a go
Factorise, then expand your answer to check it.
- 1x² + 7x + 10
- 2x² − 9x + 20
- 3x² + 2x − 15
- 4x² − 16 (hint: difference of two squares — b is 0)
- 5Solve x² + x − 20 = 0
Step 5 · Remember it
Recall
Quick quiz
Use the multiply-and-add rule.
0 of 5 answered
1.Factorise x² + 8x + 15
2.Factorise x² − 6x + 8
3.Factorise x² + 3x − 10
4.Solve (x − 7)(x + 2) = 0
5.Factorise x² − 25
Step 6 · Link it to your world
Reflect
Think about it
Quadratics describe the path of a thrown ball, the shape of bridges and profit curves.
Which part of factorising do you find hardest — finding the pair, or getting the signs right? What will you do to check next time?
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