# How do you use the discriminant to determine the nature of the roots for x^2 + 2x + 5 = 0?

Jun 19, 2015

As color(red)(Delta = -16(less than zero), this equation has two complex roots.

#### Explanation:

${x}^{2} + 2 x + 5 = 0$

The equation is of the form color(blue)(ax^2+bx+c=0 where:
$a = 1 , b = 2 , c = 5$

The Discriminant is given by:
$\Delta = {b}^{2} - 4 \cdot a \cdot c$
$= {\left(2\right)}^{2} - \left(4 \cdot \left(1\right) \cdot 5\right)$
$= 4 - 20 = - 16$

When, $\Delta < 0$ there are two complex solutions.
Here, $\textcolor{red}{\Delta = - 16}$, so this equation has two complex roots

• Note :
The solutions are normally found using the formula
$x = \frac{- b \pm \sqrt{\Delta}}{2 \cdot a}$
Finding the solutions:
 x =( -2+-sqrt-16)/(2a
color(red)( x =( -2+4i)/2 and color(red)(x = (-2-4i)/2