The quadratic formula

Notes

By now, we have seen that quadratics can be solved by factorising or completing the square. There is a third way, and that is called the quadratic formula. Basically, we solve the general equation

a x^{2} + b x + c = 0

by completing the square, and we get the formula

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

This is a formula we should memorise as it is very useful for solving hard quadratics quickly.

Proof of formula

We complete the square on the equation $a x^{2} + b x + c = 0$ to find the general formula for the roots

\begin{aligned} a x^{2} + b x + c & = 0 \\ a [x^{2} + \frac{b}{a} x] + c & = 0 \\ a [(x + \frac{b}{2 a}) - \frac{b^{2}}{4 a^{2}}] + c & = 0 \\ a (x + \frac{b}{2 a}) - \frac{b^{2}}{4 a} + c & = 0 \\ a (x + \frac{b}{2 a}) & = \frac{b^{2}}{4 a} - c \\ a (x + \frac{b}{2 a}) & = \frac{b^{2} - 4 a c}{4 a} \\ (x + \frac{b}{2 a}) & = \frac{b^{2} - 4 a c}{4 a^{2}} \\ x + \frac{b}{2 a} & = \pm \sqrt{\frac{b^{2} - 4 a c}{4 a^{2}}} \\ x & = - \frac{b}{2 a} \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a} \\ x & = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{aligned}

Example

Solve the following equations using the quadratic formula:

$2 x^{2} - x - 6 = 0$
$x^{2} + 2 - 6 x = 0$
$x^{2} - 2 \sqrt{2} x + 2 = 0$
$x^{2} + x = - 3$

Exercise

Hint

Solution

The quadratic formula ​

Notes ​

Example ​

Exercise ​

The quadratic formula

Notes

Example

Exercise