# How do you find the value of the discriminant and determine the nature of the roots 3x² + 6x - 42 = 0?

Aug 23, 2017

See a solution process below:

#### Explanation:

For $a {x}^{2} + b x + c = 0$, the values of $x$ which are the solutions to the equation are given by:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

The discriminate is the portion of the quadratic equation within the radical: ${\textcolor{b l u e}{b}}^{2} - 4 \textcolor{red}{a} \textcolor{g r e e n}{c}$

If the discriminate is:
- Positive, you will get two real solutions
- Zero you get just ONE solution
- Negative you get complex solutions

To find the discriminant for this problem substitute:

$\textcolor{red}{3}$ for $\textcolor{red}{a}$

$\textcolor{b l u e}{6}$ for $\textcolor{b l u e}{b}$

$\textcolor{g r e e n}{- 42}$ for $\textcolor{g r e e n}{c}$

${\textcolor{b l u e}{6}}^{2} - \left(4 \cdot \textcolor{red}{3} \cdot \textcolor{g r e e n}{- 42}\right) \implies$

$36 - \left(- 504\right) \implies$

$36 + 504 \implies$

$540$

Because the discriminate is positive, you will get two real solutions