How to Use the Quadratic Formula

For an equation in the form $a x^{2} + b x + c = 0$ :

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

The part under the root, $b^{2} - 4 a c$ , is the discriminant. If it's positive, there are two real solutions. If it's zero, there is one. If it's negative, there are no real solutions.

Worked examples

Solve $2 x^{2} + 3 x - 2 = 0$

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

$b^{2} - 4 a c = 9 + 16 = 25$

$x = \frac{- 3 \pm 5}{4}$ , so $x = \frac{1}{2}$ or $x = - 2$

Solve $x^{2} - 4 x + 1 = 0$

$b^{2} - 4 a c = 16 - 4 = 12$

$x = \frac{4 \pm 12}{2} = 2 \pm 3$

Common mistake

Using $b = 4$ in $x^{2} - 4 x + 1 = 0$ .

The sign is part of the coefficient, so $b = - 4$ . Then $- b = 4$ .

Try it yourself

$x^{2} - 5 x + 6 = 0$
Show answer
$x = 2$ or $x = 3$
$2 x^{2} - 4 x - 6 = 0$
Show answer
$x = 3$ or $x = - 1$
$x^{2} - 6 x + 9 = 0$
Show answer
$x = 3$

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