# How do you find the value of the discriminant and state the type of solutions given 9n^2-3n-8=-10?

Aug 8, 2016

$\Delta = - 63$
roots are not real.

#### Explanation:

The $\textcolor{b l u e}{\text{discriminant}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\Delta = {b}^{2} - 4 a c} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where a, b and c are the coefficients in the standard quadratic equation. The value of the discriminant provides information on the nature of the roots.

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}} \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{a {x}^{2} + b x + c = 0} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

•b^2-4ac>0tocolor(blue)"roots are real and irrational"

•b^2-4ac>0" and a square"tocolor(blue)"roots are real and rational"

•b^2-4ac=0tocolor(blue)"roots are real/rational and equal"

•b^2-4ac<0tocolor(blue)"roots are not real"

Equate the given equation to zero.

$\Rightarrow 9 {n}^{2} - 3 n + 2 = 0$

here a = 9 ,b =- 3 and c=2

$\Rightarrow {b}^{2} - 4 a c = {\left(- 3\right)}^{2} - \left(4 \times 9 \times 2\right) = - 63 < 0$

Since discriminant is less than zero then roots of the quadratic equation are not real.

Solving the equation using the $\textcolor{red}{\text{quadratic formula}}$

x=(3±sqrt(-63))/18=3/18±(3isqrt7)/18=1/6+-1/6isqrt7

The roots of the equation are not real. They are complex.