Sheet № 137 · Higher only · AQA · Edexcel · OCR
Factorising Harder Quadratics –
On the Higher paper you will meet quadratic expressions where the coefficient of x² is not 1, such as 6x² + 11x − 10. These are sometimes called non-monic quadratics and they require a more systematic approach than simple inspection. The most reliable method taught at GCSE is the ac method (also called splitting the middle term). This pag
§Key definitions
Question:
Factorise 2x² + 7x + 3.
Answer:
(2x + 1)(x + 3)
Q1 (Foundation):
Factorise 2x² + 5x + 2.
Q2 (Higher):
Factorise 5x² − 13x − 6.
Q3 (Higher):
Solve 4x² + 4x − 3 = 0.
§Formulas to memorise
For ax² + bx + c, find two numbers that multiply to ac and add to b
Then split bx into those two parts and factorise by grouping
Write the quadratic in the form ax² + bx + c. — Identify a, b, and c.
Calculate the product ac. — 3. Find two numbers that multiply to ac and add to b. List factor pairs of ac systematically.
Rewrite the middle term (bx) as the sum of two terms — using those numbers.
Group the four terms into two pairs — and factorise each pair.
Take out the common bracket — to complete the factorisation.
Expand to check — your answer matches the original expression.
Worked example
Factorise 2x² + 7x + 3.
Working:
⚠ Common mistakes
- ✗Using the wrong product. Students sometimes look for numbers that multiply to c instead of ac. Always calculate ac first.
- ✗Sign errors when splitting the middle term. Be very careful with negatives. If ac is negative, one of your two numbers must be positive and the other negative.
- ✗Incorrect grouping. When factorising by grouping, both groups must produce the same bracket. If they do not, check your splitting step.
✦ Exam tips
- →Write out factor pairs of ac methodically — do not guess. This avoids wasting time.
- →If you cannot find a pair, double-check your values of a, b, and c, especially signs.
- →The ac method always works for factorisable quadratics, so it is the most reliable approach.
- →After factorising, the question may ask you to solve the equation — set each bracket equal to zero.