Solving Quadratic Equations 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 must be zero. On this page you will learn how to rearrange a quadratic into the correct form, factorise it, and find both solutions. We also cover non-monic quadratics (where the coefficient of x² is not 1), which appear at Higher level. For more background on factorising itself, see our guide on Factorising Expressions.

What Is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. The highest power of x is 2, which means a quadratic can have up to two solutions (also called roots).

If (x + p)(x + q) = 0, then x + p = 0 or x + q = 0, giving x = −p or x = −q

This is called the null factor law (or zero product property). It only works when one side of the equation equals zero, which is why the first step is always to rearrange so that everything is on one side.

ax² + bx + c = 0 → factorise → set each bracket to zero → solve

Step-by-Step Method

Monic quadratics (a = 1)

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

Rearrange to ax² + bx + c = 0.
Multiply a × c.
Find two numbers that multiply to ac and add to b.
Split the middle term using these two numbers.
Factorise in pairs (grouping method).
Write as a product of two brackets and solve as above.

Quadratics with no constant term

If c = 0, for example x² − 5x = 0, simply take out x as a common factor: x(x − 5) = 0, giving x = 0 or x = 5.

Worked Example 1 — Foundation Level

Question: Solve x² + 3x − 10 = 0.

Working:

Step 1: The equation is already in the form x² + bx + c = 0.

Step 2: Find two numbers that multiply to −10 and add to 3. Pairs: (5, −2) → 5 + (−2) = 3 ✓

Step 3: Factorise: (x + 5)(x − 2) = 0.

Step 4: Set each bracket to zero. x + 5 = 0 → x = −5 x − 2 = 0 → x = 2

Check: x = 2: (2)² + 3(2) − 10 = 4 + 6 − 10 = 0 ✓ x = −5: (−5)² + 3(−5) − 10 = 25 − 15 − 10 = 0 ✓

Answer: x = −5 or x = 2

Worked Example 2 — Higher Level

Question: Solve 2x² + 7x + 3 = 0.

Working:

Step 1: a = 2, b = 7, c = 3. Multiply a × c = 6.

Step 2: Find two numbers that multiply to 6 and add to 7. Pairs: (6, 1) → 6 + 1 = 7 ✓

Step 3: Split the middle term: 2x² + 6x + x + 3 = 0.

Step 4: Factorise in pairs. 2x(x + 3) + 1(x + 3) = 0

Step 5: Factor out the common bracket. (2x + 1)(x + 3) = 0

Step 6: Solve. 2x + 1 = 0 → x = −½ x + 3 = 0 → x = −3

Check: x = −½: 2(¼) + 7(−½) + 3 = ½ − 3½ + 3 = 0 ✓

Answer: x = −½ or x = −3

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.

For a deeper dive into all quadratic methods, see our blog post on How to Solve Quadratic Equations at GCSE.

Practice Questions

Q1 (Foundation): Solve x² − 9x + 20 = 0.

Answer: x = 4 or x = 5

Q2 (Foundation/Higher): Solve x² + x − 30 = 0.

Answer: x = 5 or x = −6

Q3 (Higher): Solve 3x² − 11x − 4 = 0.

Answer: x = 4 or x = −⅓

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Summary

A quadratic equation has the form ax² + bx + c = 0 and can have up to two solutions.
Always rearrange so one side equals zero before factorising.
For monic quadratics (a = 1), find two numbers that multiply to c and add to b.
For non-monic quadratics (a ≠ 1), use the grouping (ac) method.
Apply the null factor law: set each bracket to zero to find the solutions.
Always check by substituting your solutions back into the original equation.
If the quadratic does not factorise neatly, switch to the quadratic formula or completing the square.

Solving Quadratics by Factorising –