Quadratic equations

1 Quadratic equations

Key Point 3

A quadratic equation is one which can be written in the form

$a x^{2} + b x + c = 0 a \neq 0$

where $a$ , $b$ and $c$ are given numbers and $x$ is the unknown whose value(s) must be found.

For example

$2 x^{2} + 7 x - 3 = 0, x^{2} + x + 1 = 0, 0.5 x^{2} + 3 x + 9 = 0$

are all quadratic equations. To ensure the presence of the $x^{2}$ term, the number $a$ , in the general expression $a x^{2} + b x + c$ cannot be zero. However $b$ or $c$ may be zero, so that

$4 x^{2} + 3 x = 0, 2 x^{2} - 3 = 0 and 6 x^{2} = 0$

are also quadratic equations. Frequently, quadratic equations occur in non-standard form but where necessary they can be rearranged into standard form. For example

$3 x^{2} + 5 x = 8, can be re-written as 3 x^{2} + 5 x - 8 = 0$

$2 x^{2} = 8 x - 9, can be re-written as 2 x^{2} - 8 x + 9 = 0$

$1 + x = \frac{1}{x}, can be re-written as x^{2} + x - 1 = 0$

To solve a quadratic equation we must find values of the unknown $x$ which make the left-hand and right-hand sides equal. Such values are known as solutions or roots of the quadratic equation.

Note the difference between solving quadratic equations in comparison to solving linear equations. A quadratic equation will generally have two values of $x$ (solutions) which satisfy it whereas a linear equation only has one solution.

We shall now describe three techniques for solving quadratic equations:

factorisation
completing the square
using the quadratic formula

Exercises

Verify that $x = 2$ and $x = 3$ are both solutions of $x^{2} - 5 x + 6 = 0$ .
Verify that $x = - 2$ and $x = - 3$ are both solutions of $x^{2} + 5 x + 6 = 0$ .