# What is the discriminant of x^2-2 =0 and what does that mean?

Jul 22, 2015

The discriminant of ${x}^{2} - 2 = 0$ is 8,
which means there are 2 Real solutions to this equation.

#### Explanation:

For a quadratic equation in the standard form
$\textcolor{w h i t e}{\text{XXXX}}$$a {x}^{2} + b x + c = 0$
the discriminant is
$\textcolor{w h i t e}{\text{XXXX}}$$\Delta = {b}^{2} - 4 a c$

$\Delta \left\{\begin{matrix}< 0 & \rightarrow \text{there are no Real solutions" \\ = 0 & rarr "there is exactly 1 Real solution" \\ > 0 & rarr "there are 2 Real solutions}\end{matrix}\right.$

Converting the given equation ${x}^{2} - 2 = 0$
into standard form
$\textcolor{w h i t e}{\text{XXXX}}$$1 {x}^{2} + 0 x - 2 = 0$
gives us
$\textcolor{w h i t e}{\text{XXXX}}$$a = 1$$\textcolor{w h i t e}{\text{XXXX}}$$b = 0$$\textcolor{w h i t e}{\text{XXXX}}$$c = - 2$

So the discriminant is
$\textcolor{w h i t e}{\text{XXXX}}$$\Delta = {0}^{2} - 4 \left(1\right) \left(- 2\right) = + 8$

which implies that there are 2 Real solutions for $x$