Sheet № 19 · Foundation + Higher · AQA · Edexcel · OCR
Solving Quadratics by Factorising –
Solving quadratic equations by factorising is one of the most heavily examined algebra topics at GCSE. It appears on both Foundation and Higher tier papers, and it is the quickest method when the quadratic factorises neatly. The approach relies on a simple but powerful principle: if two things multiply to give zero, at least one of them m
§Key definitions
Question:
Solve x² + 3x − 10 = 0.
Check:
x = 2: (2)² + 3(2) − 10 = 4 + 6 − 10 = 0 ✓
Answer:
x = −5 or x = 2
Q1 (Foundation):
Solve x² − 9x + 20 = 0.
Q2 (Foundation/Higher):
Solve x² + x − 30 = 0.
§Formulas to memorise
If (x + p)(x + q) = 0, then x + p = 0 or x + q = 0, giving x = −p or x = −q
ax² + bx + c = 0 → factorise → set each bracket to zero → solve
Rearrange — the equation so that one side is zero: x² + bx + c = 0.
Find two numbers — that multiply to c and add to b.
Write the factorised form: — (x + p)(x + q) = 0.
Set each bracket equal to zero: — x + p = 0 → x = −p; x + q = 0 → x = −q.
State both solutions. — ### Non-monic quadratics (a ≠ 1) — Higher tier
Multiply a × c. — 3. Find two numbers that multiply to ac and add to b.
Split the middle term — using these two numbers.
Factorise in pairs — (grouping method).
Rearrange to ax² + bx + c = 0.
Worked example
Solve x² + 3x − 10 = 0.
Working:
⚠ Common mistakes
- ✗Forgetting to rearrange to zero first. If the equation is x² + 3x = 10, you must subtract 10 from both sides before factorising. You cannot factorise when one side is not zero.
- ✗Only finding one solution. A quadratic has up to two solutions. Even if both brackets look similar, set each one to zero individually.
- ✗Sign errors when listing factor pairs. With negative values of c, one number will be positive and one negative. List all pairs systematically.
- ✗Forgetting x = 0 as a solution. In x² − 7x = 0, factorising gives x(x − 7) = 0. Many students find x = 7 but forget x = 0.
- ✗Confusing this method with the quadratic formula. Factorising is preferred when the quadratic factorises cleanly. If you cannot find integer factor pairs, use the quadratic formula instead.
✦ Exam tips
- →Check whether the equation is already set to zero. If not, rearrange first — this is worth a mark on most papers.
- →Write "= 0" at the end of your factorised expression. Without it, you are writing a factorisation, not solving an equation.
- →On Foundation papers, quadratics almost always have integer solutions. If you are getting fractions, double-check your factor pairs.
- →For non-monic quadratics, practise the grouping method until it feels natural. It appears on Higher papers most years across all exam boards.