To solve a quadratic equation by factorisation, make one side zero, factorise the other side, then set each factor equal to zero. Each factor gives one root.
This lesson follows recognising a quadratic from its highest power and is part of SPM Mathematics quadratic functions and equations.
What are the three steps?
- Rearrange so that one side is 0, in the form ax² + bx + c = 0.
- Factorise the quadratic.
- Set each bracket to 0 and solve.
The logic rests on one fact: if A × B = 0, then A = 0 or B = 0. This is why step 1 is not optional.
Worked example: a simple case
Solve x² − 7x + 12 = 0. Look for two numbers that multiply to 12 and add to −7. They are −3 and −4.
(x − 3)(x − 4) = 0, so x − 3 = 0 or x − 4 = 0. The roots are x = 3 and x = 4.
Check in the original: 3² − 7(3) + 12 = 9 − 21 + 12 = 0, and 4² − 7(4) + 12 = 16 − 28 + 12 = 0.
Worked example: numbers on both sides
Solve x² − 5x = 14. Do not factorise the left side yet. Subtract 14 from both sides first: x² − 5x − 14 = 0.
Now find two numbers that multiply to −14 and add to −5. They are −7 and 2. So (x − 7)(x + 2) = 0, and x = 7 or x = −2.
Check: 7² − 5(7) = 49 − 35 = 14, and (−2)² − 5(−2) = 4 + 10 = 14.
Two mistakes that cost marks
Mistake 1: factorising without a zero. The student writes x(x − 5) = 14 and then says x = 14 or x − 5 = 14, which gives x = 14 or 19. Test x = 14: 14 × 9 = 126, not 14, so the answer fails.
Mistake 2: dividing out x. For 3x² = 12x, dividing by x gives 3x = 12 and x = 4. The root x = 0 has vanished.
| Equation | Wrong method | Right method |
|---|---|---|
| x² − 5x = 14 | x(x − 5) = 14, so x = 14 | x² − 5x − 14 = 0, so x = 7 or −2 |
| 3x² = 12x | Divide by x, so x = 4 | 3x² − 12x = 0, so 3x(x − 4) = 0, roots 0 and 4 |
When the coefficient of x² is not 1
Solve 2x² + 7x − 4 = 0. Multiply a and c to get 2 × (−4) = −8.
Find two numbers that multiply to −8 and add to 7. They are 8 and −1.
Split the middle term: 2x² + 8x − x − 4 = 0. Group: 2x(x + 4) − 1(x + 4) = 0, so (2x − 1)(x + 4) = 0.
The roots are x = ½ and x = −4. Check x = ½: 2(¼) + 3.5 − 4 = 0.5 + 3.5 − 4 = 0.
Check yourself
Solve (a) 3x² = 12x and (b) x² + 2x = 15.
Answer
(a) 3x² − 12x = 0, so 3x(x − 4) = 0. The roots are x = 0 and x = 4.
(b) x² + 2x − 15 = 0. The numbers 5 and −3 multiply to −15 and add to 2, so (x + 5)(x − 3) = 0.
The roots are x = −5 and x = 3. Check x = 3: 9 + 6 = 15.
What to study next
The roots you found are the x-values where the graph of the function touches the x-axis. Continue with finding roots from a quadratic graph, or test yourself on the quadratic practice set.
If factorising breaks down when the layout changes, online one-to-one Mathematics tuition lets a teacher rebuild it with you step by step.