Quadratic Formula

Exam code: 4037

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

Quadratic formula

A quadratic equation has the form:
ax² + bx + c = 0 (as long as a ≠ 0)
- you need "= 0" on one side
The quadratic formula is a formula that gives both solutions:
- $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

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 $2 x^{2} - 5 x + 2 = 0$ using the quadratic formula:
- a = 2, b = -5 and c = 2
- $x = \frac{- (- 5) \pm \sqrt{{(- 5)}^{2} - 4 \times 2 \times 2}}{2 \times 2}$
- Carefully simplify by doing the calculation in parts $x = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm \sqrt{9}}{4}$
- Separate the + and - to get the two answers $\begin{array}{rcl} x & = & \frac{5 + 3}{4} = \frac{8}{4} = 2 \\ x & = & \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2} \end{array}$
On the non-calculator paper, answers may be required in exact form
On the calculator paper, you might have to round your answers
- Give your answers in exact form before rounding in case you round incorrectly

Worked example

Use the quadratic formula, without a calculator, to find the exact solutions of the equation $12 x^{2} - 17 x + 6 = 0$ .

Write down the values of a, b and c

$a = 12, b = - 17, c = 6$

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

$x = \frac{- (- 17) \pm \sqrt{{(- 17)}^{2} - 4 \times 12 \times 6}}{2 \times 12}$

Simplify

$\begin{array}{rcl} x & = & \frac{17 \pm \sqrt{289 - 288}}{24} \\ = & \frac{17 \pm \sqrt{1}}{24} \end{array}$

Work out the + and - parts separately to get the two solutions

$x = \frac{17 + 1}{24} = \frac{18}{24} x = \frac{17 - 1}{24} = \frac{16}{24}$

$x = \frac{3}{4}, x = \frac{2}{3}$