Watch out
Forgetting to make the equation equal zero.
Back to National 5 Mathematics
SQA National 5 Mathematics
Solving quadratic equations by factorising
Finding roots from factorised form.
Formula reference
Check the National 5 rules and formulae linked to this topic.
Which method?
Match exam clues to a suitable method.
Before you start
- Make one side equal to zero.
- Factorise the quadratic.
- Use the zero-product rule.
Explanation
To solve a quadratic by factorising, write it as two brackets equal to zero.
If two factors multiply to zero, at least one factor must be zero.
This gives two possible solutions unless the brackets are repeated.
Key formulae and rules
- If (x − a)(x − b) = 0, then x = a or x = b
Check
Substitute each solution back into the equation. A quick check catches most sign errors.
Exam tip
The zero-product rule only works once the equation is equal to zero.
Calculator tip
Enter the numerator in brackets, especially when using a negative b or the ± answers separately.
Worked examples
Worked example 1
Two positive roots
Solve x² − 5x + 6 = 0.
- Factorise: (x − 2)(x − 3) = 0
- Set each bracket to zero.
Answer: x = 2 or x = 3
Worked example 2
One negative root
Solve x² + 2x − 15 = 0.
- Factorise: (x + 5)(x − 3) = 0
- x + 5 = 0 or x − 3 = 0
So: x = −5 or x = 3
Worked example 3
Common factor first
Solve 3x² − 12x = 0.
- Factorise: 3x(x − 4) = 0
- 3x = 0 or x − 4 = 0
Answer: x = 0 or x = 4
Watch out
- Forgetting to make the equation equal zero.
- Stopping after factorising and not solving.
- Missing x = 0 when there is a common factor