# How do you find the discriminant of 3x^2+2x-1=0 and use it to determine if the equation has one, two real or two imaginary roots?

Apr 12, 2017

There are 2 real roots
(see below)

#### Explanation:

For a quadratic with the general form:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{a} {x}^{2} + \textcolor{b l u e}{b} x + \textcolor{g r e e n}{c} = 0$
the discriminant is:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{\mathmr{and} a n \ge}{\Delta} = {\textcolor{b l u e}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{g r e e n}{c}$
with
color(white)("XXX")color(orange)(Delta){(color(orange)( < 0) rarr 2" imaginary roots"),(color(orange)(= 0) rarr 1 " real root"),(color(orange)(> 0) rarr 2 " real roots"):}

Given
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{3} {x}^{2} + \textcolor{b l u e}{2} x + \textcolor{g r e e n}{\left(- 1\right)} = 0$
the discriminant is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{\mathmr{and} a n \ge}{\Delta} = {\textcolor{b l u e}{2}}^{2} - 4 \cdot \textcolor{red}{3} \cdot \textcolor{g r e e n}{\left(- 1\right)} = \textcolor{\mathmr{and} a n \ge}{20}$
$\Rightarrow$ there are 2 real roots.