# The quadratic formula

A formula that solves any quadratic ax^2 + bx + c = 0, even when it does not factorise easily.

Maths (school) · Part 03: Equations and formulae · Card 7 of 8 · Last updated 2026-10-19

## Analogy

Like a master key that opens every lock, even when you cannot pick it.

## Example

**2x^2 + 3x - 2 = 0**

a = 2, b = 3, c = -2
x = (-3 +/- sqrt(9 + 16)) / 4
x = (-3 +/- 5) / 4
x = 0.5 or x = -2

## In practice

Used to find landing points when numbers are not tidy, such as in projectile problems.

## Practice

### Flashcards

- **What is the quadratic formula?** x = (-b +/- sqrt(b^2 - 4ac)) / 2a.

### Interview questions

**Solve 2x^2 + 3x - 2 = 0 using the quadratic formula.**

- a = 2, b = 3, c = -2
- Discriminant = 9 + 16 = 25, square root 5
- x = (-3 + 5) / 4 = 0.5
- x = (-3 - 5) / 4 = -2

## Questions people ask

### What is the quadratic formula?

It is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. You read a, b and c from the equation, put them in, and work out both answers. The plus gives one solution and the minus gives the other. Take care with signs: if b is negative, then -b becomes positive. Write out a, b and c before substituting.

### What does the part under the square root tell you?

The expression b^2 - 4ac is called the discriminant. If it is positive, there are two answers. If it is zero, there is one. If it is negative, there are no real answers, because you cannot square root a negative.

## Related concepts

- [Solving quadratics by factorising](https://studyvisually.com/learn/science/maths-school/maths-school-part-3/maths-school-part-3-concept-6)
- [Solving linear equations](https://studyvisually.com/learn/science/maths-school/maths-school-part-3/maths-school-part-3-concept-1)
- [Unknowns on both sides](https://studyvisually.com/learn/science/maths-school/maths-school-part-3/maths-school-part-3-concept-2)
- [Changing the subject](https://studyvisually.com/learn/science/maths-school/maths-school-part-3/maths-school-part-3-concept-3)

Web page: https://studyvisually.com/learn/science/maths-school/maths-school-part-3/maths-school-part-3-concept-7
Study this card in the app: https://studyvisually.com/app/learn?category=maths-school&card=maths-school-part-3-concept-7