# How do you find the value of the discriminant and determine the nature of the roots -9b^2=-8b+8?

Aug 22, 2017

See a solution process below:

#### Explanation:

First, convert the equation to standard form:

$- 9 {b}^{2} + \textcolor{red}{8 b} - \textcolor{b l u e}{8} = - 8 b + 8 + \textcolor{red}{8 b} - \textcolor{b l u e}{8}$

$- 9 {b}^{2} + 8 b - 8 = 0$

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 $\textcolor{red}{a}$

$\textcolor{b l u e}{8}$ for $\textcolor{b l u e}{b}$

$\textcolor{g r e e n}{- 8}$ for $\textcolor{g r e e n}{c}$

${\textcolor{b l u e}{8}}^{2} - \left(4 \cdot \textcolor{red}{- 9} \cdot \textcolor{g r e e n}{- 8}\right)$

$64 - 288$

$- 224$

Because the discriminate is negative there will be a complex solution.