# How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given -3x^2-5x+2=0?

Nov 23, 2016

The solution is $S = \left\{- 2 , \frac{1}{3}\right\}$

#### Explanation:

The equation is $- 3 {x}^{2} - 5 x + 2 = 0$

The quadratic equation is $a {x}^{2} + b x + c = 0$

We start by calculating the discriminant

$\Delta = {b}^{2} - 4 a c = {\left(- 5\right)}^{2} - 4 \left(- 3\right) \left(+ 2\right)$
$= 25 + 27 = 49$

As, $\Delta > 0$, we expect 2 real roots

The roots are
$= \frac{- b \pm \sqrt{\Delta}}{2 a}$

${x}_{1} = \frac{5 + 7}{- 6} = - 2$

${x}_{2} = \frac{5 - 7}{- 6} = \frac{1}{3}$