# Factorising quadratics

Writing an expression like x^2 + 7x + 12 as two brackets multiplied together.

Maths (school) · Part 02: Algebra basics · Card 6 of 8 · Last updated 2026-10-19

## Analogy

Like finding two numbers that fit a lock: they must multiply to one number and add to another.

## Example

**x^2 + 7x + 12**

Need two numbers: multiply to 12, add to 7
3 and 4 work
= (x + 3)(x + 4)

## In practice

Engineers factorise quadratics to find where a thrown ball lands.

## Practice

### Flashcards

- **Factorise x^2 + 7x + 12.** (x + 3)(x + 4). 3 x 4 = 12 and 3 + 4 = 7.

### Interview questions

**Factorise x^2 + 5x + 6 and explain how you know your answer is right.**

- Find two numbers that multiply to 6 and add to 5
- They are 2 and 3
- Write (x + 2)(x + 3)
- Expand to check: x^2 + 3x + 2x + 6

## Questions people ask

### What do the two numbers need to do?

They must multiply to give the last number and add to give the middle number. For x^2 + 7x + 12, 3 x 4 = 12 and 3 + 4 = 7. So the brackets are (x + 3)(x + 4). Try factor pairs of the last number until the sum fits.

### What if the last number is negative?

Then the two numbers have opposite signs, one positive and one negative. For x^2 + x - 12, you need numbers that multiply to -12 and add to 1. These are 4 and -3, so the answer is (x + 4)(x - 3).

## Related concepts

- [Difference of two squares](https://studyvisually.com/learn/science/maths-school/maths-school-part-2/maths-school-part-2-concept-7)
- [Simplifying expressions](https://studyvisually.com/learn/science/maths-school/maths-school-part-2/maths-school-part-2-concept-1)
- [Substitution](https://studyvisually.com/learn/science/maths-school/maths-school-part-2/maths-school-part-2-concept-2)
- [Expanding brackets](https://studyvisually.com/learn/science/maths-school/maths-school-part-2/maths-school-part-2-concept-3)

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