Watch out
Finding numbers that add to c instead of multiply to c.
Back to National 5 Mathematics
SQA National 5 Mathematics
Factorising quadratics
Writing quadratic trinomials as two brackets.
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
- Put the quadratic in ax² + bx + c form
- Find factors of c.
- Check the middle term after expanding.
Explanation
Factorising quadratics reverses expanding double brackets.
For x² + bx + c, find two numbers that multiply to c and add to b.
If the coefficient of x² is not 1, check factor pairs more carefully by expanding.
Key formulae and rules
- x² + bx + c = (x + p)(x + q), where p + q = b and pq = c
Check
Substitute each solution back into the equation. A quick check catches most sign errors.
Exam tip
Always expand mentally or in writing to check the middle term.
Calculator tip
Enter the numerator in brackets, especially when using a negative b or the ± answers separately.
Worked examples
Worked example 1
Positive signs
Factorise x² + 7x + 12
- Find two numbers that multiply to 12 and add to 7.
- 3 and 4 work.
Answer: (x + 3)(x + 4).
Worked example 2
Negative constant
Factorise x² + x − 12
- Find two numbers that multiply to −12 and add to 1.
- 4 and −3 work.
So: (x + 4)(x − 3).
Worked example 3
Coefficient greater than 1
Factorise 2x² + 7x + 3
- Try (2x + 1)(x + 3).
- Expand to check: 2x² + 6x + x + 3
Answer: (2x + 1)(x + 3).
Watch out
- Finding numbers that add to c instead of multiply to c.
- Losing negative signs.
- Not checking by expanding.