# What is the discriminant of x^2+2x+8=0 and what does that mean?

Jul 15, 2015

The discriminant of ${x}^{2} + 2 x + 8 = 0$ is $\left(- 28\right)$ which means that this equation has no Real solutions.

#### Explanation:

For a quadratic equation in the 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$

The discriminant is the portion of the quadratic formula for solving a quadratic equation:
$\textcolor{w h i t e}{\text{XXXX}}$$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Seen in this context, it should be clear why:
$\textcolor{w h i t e}{\text{XXXX}}$Delta { ( > 0, rarr, 2 " Real solutions"), (=0,rarr, 1 " Real solution"), (< 0, rarr, " no Real solutions"):}

$\textcolor{w h i t e}{\text{XXXX}}$${x}^{2} + 2 x + 8 = 0$
$\textcolor{w h i t e}{\text{XXXX}}$$\Delta = {2}^{2} - 4 \left(1\right) \left(8\right) = - 28$