# How do you find the discriminant for 9x^2+24x=-16 and determine the number and type of solutions?

Apr 12, 2018

See a solution process below:

#### Explanation:

First, write the equation in standard form:

$9 {x}^{2} + 24 x + \textcolor{red}{16} = - 16 + \textcolor{red}{16}$

$9 {x}^{2} + 24 x + 16 = 0$

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:

$\textcolor{red}{9}$ for $\textcolor{red}{a}$

$\textcolor{b l u e}{24}$ for $\textcolor{b l u e}{b}$

$\textcolor{g r e e n}{16}$ for $\textcolor{g r e e n}{c}$

${\textcolor{b l u e}{24}}^{2} - \left(4 \cdot \textcolor{red}{9} \cdot \textcolor{g r e e n}{16}\right)$

$576 - \left(36 \cdot \textcolor{g r e e n}{16}\right)$

$576 - 576$

$0$

Because the discriminate is $0$ there is just one solution to this problem.