What if I cannot find two numbers that work?

Some quadratics do not factorise with whole numbers. If you have checked every factor pair and none adds to the middle number, use the quadratic formula or completing the square instead. In exams, a question that says "give your answer to 2 decimal places" is a strong hint that it will not factorise neatly.

Why does the equation have to equal zero before I factorise?

Because the method relies on a simple fact. If two numbers multiply to give zero, at least one of them must be zero. That only works with zero. If two brackets multiply to give 2, you cannot say either bracket equals 2.

How can I check my factorising?

Expand your brackets and see if you get back the original expression. It takes a few seconds and catches sign mistakes, which are the most common error.

Maths explained

Factorising quadratic equations step by step, made simple

How to factorise quadratics step by step: common factors, x² + bx + c, splitting the middle term when a is not 1, difference of squares, and solving.

For students. By the Qlive team. 28 September 2026 . 5 min read.

This guide is for you if you are in roughly Grades 9 to 11 and quadratics feel like guesswork. In CBSE, factorising by splitting the middle term is taught in Class 9 Polynomials and used to solve equations in Class 10 Quadratic Equations. It is also a key skill in ICSE Class 10, GCSE Higher and IGCSE Extended maths. With a clear method, it stops being guesswork.

The idea in plain words

A quadratic expression has an x² term as its highest power, like x² + 7x + 12. The general form is ax² + bx + c.

Factorising means writing it as a product of brackets, like (x + 3)(x + 4). It is the reverse of expanding brackets.

You can check this one by expanding: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12.

Notice where the numbers come from:

3 × 4 = 12 (the constant term, c)
3 + 4 = 7 (the middle number, b)

That pattern is the key to the whole topic.

Before you start: always look for a common factor

If every term shares a factor, take it out first.

3x² - 12x = 3x(x - 4)
2x² + 10x + 12 = 2(x² + 5x + 6) = 2(x + 2)(x + 3)

This makes everything after it easier.

Worked examples

Example 1: x² + bx + c with all positive terms

Factorise x² + 7x + 12.

You need two numbers that multiply to 12 and add to 7.
List the factor pairs of 12: 1 and 12 (sum 13), 2 and 6 (sum 8), 3 and 4 (sum 7).
3 and 4 work.

Answer: (x + 3)(x + 4)

Example 2: negative terms

Factorise x² - 2x - 15.

You need two numbers that multiply to -15 and add to -2.
A negative product means one number is positive and one is negative.
Try pairs: -5 and 3 multiply to -15 and add to -2.

Answer: (x - 5)(x + 3)

Check: (x - 5)(x + 3) = x² + 3x - 5x - 15 = x² - 2x - 15. Correct.

A quick guide to the signs:

c is…	b is…	The two numbers are…
positive	positive	both positive
positive	negative	both negative
negative	either	one positive, one negative (the larger one has the sign of b)

Example 3: when a is not 1 (splitting the middle term)

Factorise 2x² + 7x + 3.

Multiply a × c: 2 × 3 = 6
Find two numbers that multiply to 6 and add to 7: 6 and 1.
Split the middle term using these numbers: 2x² + 6x + x + 3
Group in pairs and factorise each pair:
- 2x² + 6x = 2x(x + 3)
- x + 3 = 1(x + 3)
The bracket (x + 3) is common: (x + 3)(2x + 1)

Answer: (2x + 1)(x + 3)

Check: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. Correct.

Example 4: difference of two squares

When you have one square minus another, there is a shortcut:

a² - b² = (a - b)(a + b)

x² - 49 = (x - 7)(x + 7)
4x² - 25 = (2x)² - 5² = (2x - 5)(2x + 5)

Example 5: solving a quadratic equation

Solve x² - 5x + 6 = 0.

Make sure one side is zero. It already is.
Factorise: two numbers that multiply to 6 and add to -5 are -2 and -3. So (x - 2)(x - 3) = 0
If two things multiply to zero, one of them must be zero:
- x - 2 = 0, so x = 2
- x - 3 = 0, so x = 3

Answer: x = 2 or x = 3

Check x = 2: 4 - 10 + 6 = 0. Check x = 3: 9 - 15 + 6 = 0. Both correct.

Common mistakes and how to avoid them

Sign errors. This is the most common mistake by far. Always expand your answer to check it.

Missing the common factor. If you factorise 2x² + 10x + 12 as (2x + 4)(x + 3), it is not wrong, but it is not fully factorised, because (2x + 4) = 2(x + 2). Take common factors out first.

Solving without making one side zero. For x² - 5x = -6, first add 6 to both sides to get x² - 5x + 6 = 0, then factorise.

Dividing by x and losing an answer. For x² = 3x, dividing by x gives x = 3 only. Instead rearrange to x² - 3x = 0, factorise to x(x - 3) = 0, and get x = 0 or x = 3.

Getting the signs of the solutions backwards. If (x + 4)(x - 2) = 0, the solutions are x = -4 and x = 2. Each solution has the opposite sign to the number in its bracket.

Thinking x² + 9 factorises. A sum of two squares does not factorise into real brackets. Only a difference does.

Try these

Factorise x² + 9x + 20.
Factorise x² - x - 12.
Factorise x² - 10x + 21.
Factorise 6x² - x - 2.
Solve x² + 2x - 8 = 0.

Answers

4 × 5 = 20 and 4 + 5 = 9: (x + 4)(x + 5).
-4 × 3 = -12 and -4 + 3 = -1: (x - 4)(x + 3).
-3 × -7 = 21 and -3 + -7 = -10: (x - 3)(x - 7).
a × c = -12. Numbers that multiply to -12 and add to -1 are -4 and 3. 6x² - 4x + 3x - 2 = 2x(3x - 2) + 1(3x - 2) = (2x + 1)(3x - 2).
(x + 4)(x - 2) = 0, so x = -4 or x = 2.

Still unsure?

Factorising leans on times tables, negative numbers and expanding brackets, so a wobble in any of those makes quadratics feel hard. A Qlive teacher can work through it with you one-to-one, starting with a free assessment that finds the exact gap. You can read how we teach on our maths page and our CBSE page.