The Quadratic Formula

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Quadratic Formula

Quadratic formula

A quadratic equation has the form ax² + bx + c = 0 (where a ≠ 0)
- you need "= 0" on one side
The quadratic formula is a formula that gives both solutions to a quadratic equation:

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Read off the values of a, b and c from the equation
Substitute these into the formula
- Write this line of working in the exam
- Put brackets around any negative numbers being substituted in
To solve 2x² - 8x - 3 = 0 using the quadratic formula:
- a = 2, b = -8 and c = -3
- $x = \frac{- (- 8) \pm \sqrt{{(- 8)}^{2} - 4 \times 2 \times (- 3)}}{2 \times 2}$
- Either type this into a calculator or simplify by hand
  - Type it once using + for ± then again using - for ±
- The solutions are x = 4.3452078... or x = -0.34520787....
  - To 3 decimal places: x = 4.345 or x = -0.345
  - To 3 significant figures: x = 4.35 or x = -0.345

You may be asked to give answers in an exact (surd) form
In the example above, work out the number under the square root sign
- Be careful with negatives!
  - ${(- 8)}^{2} - 4 \times 2 \times (- 3) = 64 + 24 = 88$
- Now square root this number and use surd rules to simplify
  - $\sqrt{88} = \sqrt{4 \times 22} = \sqrt{4} \times \sqrt{22} = 2 \sqrt{22}$
- Substitute this back into the formula and simplify
  - $x = \frac{8 \pm 2 \sqrt{22}}{4} = \frac{2 (4 \pm \sqrt{22})}{4} = \frac{4 \pm \sqrt{22}}{2}$
  - The solutions in exact (surd) form are $x = \frac{4 + \sqrt{22}}{2}$ or $x = \frac{4 - \sqrt{22}}{2}$
Calculators that can solve quadratics will give solutions in exact (surd) form

The part of the formula under the square root (b² – 4ac) is called the discriminant
The sign of this value tells you if there are 0, 1 or 2 solutions
- If b² – 4ac > 0 (positive)
  - then there are 2 different solutions
- If b² – 4ac = 0
  - then there is only 1 solution
- If b² – 4ac < 0 (negative)
  - then there are no real solutions
If the discriminant is a perfect square number (0, 1, 4, 9, 16, …) then the quadratic expression can be factorised using integers

If your calculator solves quadratic equations, use it to check your final answers
- But a correct method and working must still be shown

Worked example

Use the quadratic formula to find the solutions of the equation 3x² - 2x - 4 = 0.
Give each solution as an exact value in its simplest form.

Answer:

Write down the values of a, b and c

a = 3, b = -2, c = -4

Substitute these values into the quadratic formula, $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
Put brackets around any negative numbers

$x = \frac{- (- 2) \pm \sqrt{{(- 2)}^{2} - 4 \times 3 \times (- 4)}}{2 \times 3}$

Simplify the expressions

$x = \frac{2 \pm \sqrt{4 + 48}}{6} = \frac{2 \pm \sqrt{52}}{6}$

Simplify the surd

$x = = \frac{2 \pm \sqrt{4 \times 13}}{6} = \frac{2 \pm 2 \sqrt{13}}{6}$

Simplify the fraction

$x = \frac{1 \pm \sqrt{13}}{3}$

$x = \frac{1 + \sqrt{13}}{3} or x = \frac{1 - \sqrt{13}}{3}$