How to Solve Quadratic Equations by Factorising: A Step-by-Step GCSE Guide

Quadratic equations are one of the biggest scoring opportunities on the Higher tier — and a regular visitor on Foundation too. The good news is that almost every quadratic you’ll meet at GCSE can be cracked with one of three factorising methods, and when factorising fails, the quadratic formula is there as a guaranteed backup. This guide walks through all of it with worked examples, in the order you should try them.

Start by setting the equation to zero

Every method below relies on the same first principle: a quadratic equation must equal zero before you factorise it. Once you’ve factorised, you set each bracket equal to zero separately.

Why does this work? If two things multiply to give zero, at least one of them must be zero — because zero times anything is zero. So if (x + 7)(x − 7) = 0, then either x + 7 = 0 or x − 7 = 0. That single idea is the engine behind solving by factorising.

Method 1: The difference of two squares

When you see one square subtracted from another — and no middle term — reach for the difference of two squares. The pattern is a² − b² = (a + b)(a − b).

Take x² − 49 = 0. The square root of x² is x, and the square root of 49 is 7, so it factorises straight to (x + 7)(x − 7) = 0. Setting each bracket to zero gives x = −7 or x = 7.

This works even when the x² has a coefficient that is itself a perfect square. For 9x² − 64 = 0, the square root of 9x² is 3x and the square root of 64 is 8, so it becomes (3x + 8)(3x − 8) = 0, giving x = −8/3 or x = 8/3.

Tip: If the numbers aren’t perfect squares, look for a common factor first. 3x² − 75 = 0 doesn’t fit the pattern — until you take out the 3 to get 3(x² − 25) = 0. Now x² − 25 is a difference of two squares: 3(x + 5)(x − 5) = 0, so x = −5 or x = 5.

Method 2: Trinomials where the x² coefficient is 1

When you have a full three-term quadratic like x² − 2x − 15 = 0, find two numbers that multiply to give the constant term and add to give the middle coefficient.

Here you need two numbers that multiply to −15 and add to −2. The factor pairs of 15 are 5 and 3. Testing the signs: −5 + 3 = −2 ✓ (and −5 × 3 = −15 ✓). So the equation factorises to (x − 5)(x + 3) = 0, giving x = 5 or x = −3.

Let’s try x² + 3x − 28 = 0. You need two numbers that multiply to −28 and add to +3. Run through the factor pairs of 28 — 1 and 28, 2 and 14, 4 and 7. The pair 4 and 7 differ by 3, so with the right signs it’s −4 and +7: that gives (x − 4)(x + 7) = 0, so x = 4 or x = −7.

Did you know? Listing factor pairs systematically is faster than guessing. Once you find a pair whose difference or sum matches the middle number, you only need to sort out the signs — which is exactly the kind of disciplined approach we cover in How to Get a Grade 9 in GCSE Maths.

Method 3: When the x² coefficient is not 1

This is the case students fear most, but it follows a fixed recipe. Take 8x² + 2x − 15 = 0.

1. Multiply the first and last coefficients: 8 × −15 = −120.
2. Find two numbers that multiply to −120 and add to +2. Make a list if you need to: testing pairs, 12 and −10 work (12 × −10 = −120, and 12 + (−10) = 2).
3. Split the middle term using those numbers: 8x² + 12x − 10x − 15 = 0.
4. Factorise by grouping. From the first two terms, take out 4x → 4x(2x + 3). From the last two, take out −5 → −5(2x + 3).
5. Check the brackets match. Both give (2x + 3) — that’s how you know you’re on the right track. Write it once, and put the outside terms in the second bracket: (2x + 3)(4x − 5) = 0.

Setting each bracket to zero: 2x + 3 = 0 gives x = −3/2, and 4x − 5 = 0 gives x = 5/4.

Here’s a summary of every worked example so you can see the methods side by side:

Equation	Method	Factorised form	Solutions
x² − 49 = 0	Difference of two squares	(x + 7)(x − 7)	x = 7 or x = −7
3x² − 75 = 0	Common factor, then DOTS	3(x + 5)(x − 5)	x = 5 or x = −5
9x² − 64 = 0	Difference of two squares	(3x + 8)(3x − 8)	x = 8/3 or x = −8/3
x² − 2x − 15 = 0	Two numbers (×−15, +−2)	(x − 5)(x + 3)	x = 5 or x = −3
x² + 3x − 28 = 0	Two numbers (×−28, +3)	(x − 4)(x + 7)	x = 4 or x = −7
8x² + 2x − 15 = 0	Splitting the middle term	(2x + 3)(4x − 5)	x = −3/2 or x = 5/4

The backup: the quadratic formula

When factorising won’t come — ugly numbers, or solutions that aren’t whole — the quadratic formula always delivers. For an equation in the standard form ax² + bx + c = 0:

Remember: x = (−b ± √(b² − 4ac)) ÷ 2a. On Higher tier this is not given to you, so practise writing it from memory.

Watch how it reproduces the factorising answers. For x² − 2x − 15 = 0, we have a = 1, b = −2, c = −15. Then b² − 4ac = (−2)² − 4(1)(−15) = 4 + 60 = 64, and √64 = 8. So x = (2 ± 8) ÷ 2, giving (2 + 8)/2 = 5 and (2 − 8)/2 = −3 — exactly the same as factorising.

For 8x² + 2x − 15 = 0, a = 8, b = 2, c = −15. Then b² − 4ac = 4 + 480 = 484, and √484 = 22. So x = (−2 ± 22) ÷ 16, giving 20/16 = 5/4 and −24/16 = −3/2. Identical again.

Tip: The most common formula mistakes are sign errors on b and forgetting that −4ac becomes positive when c is negative. Slow down on those two steps. For more guidance on protecting marks under pressure, see The Non-Calculator Paper: 10 Tactics That Actually Save Marks.

Which method should you reach for?

Use this quick decision order in the exam:

• Two terms, both squares? Difference of two squares.
• Three terms, x² coefficient of 1? Find two numbers that multiply and add correctly.
• Three terms, x² coefficient more than 1? Split the middle term and factorise by grouping.
• Stuck after 30 seconds, or messy numbers? Quadratic formula.

If you’re still deciding which paper you’ll sit and how much of this you need, our guide to Foundation vs Higher Tier breaks down exactly where quadratics appear on each.

Practise quadratics until they’re automatic

Reading worked examples is only step one — quadratics stick when you drill them under timed conditions until the method becomes instinct. That’s what our apps are built for.

GCSE Algebra takes you through expressions, equations, inequalities, sequences and graphs across both Foundation and Higher tier, with quadratic factorising and the formula covered in depth and instant marking so you can spot exactly where your sign errors creep in. If you’d rather revise every topic in one place, GCSE Maths bundles Number, Algebra, Geometry and Statistics together, so you can move from quadratics straight into the geometry questions that combine with them — part of the 2900+ questions across the four subject apps.

Pair regular practice with the structure in our 12-Week GCSE Maths Revision Plan, and quadratics will go from being the topic you dread to one of your most reliable sources of marks. You’ve got this.