# How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given 9x^2-6x-4=-5?

Dec 1, 2016

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

#### Explanation:

Let's rewrite the quadratic equation in the form

$a {x}^{2} + b x + c = 0$

$9 {x}^{2} - 6 x - 4 + 5 = 0$

$9 {x}^{2} - 6 x + 1 = 0$

Let's calculate the discriminant

$\Delta = {b}^{2} - 4 a c = {\left(- 6\right)}^{2} - 4 \cdot 9 \cdot 1 = 36 - 36 = 0$

As, $\Delta = 0$, we have a double real root

$x = \frac{- b \pm \sqrt{\Delta}}{2 a} = - \frac{b}{2 a} = \frac{6}{18} = \frac{1}{3}$

We can also factorise the quadratic equation

$9 {x}^{2} - 6 x + 1 = \left(3 x - 1\right) \left(3 x - 1\right) = {\left(3 x - 1\right)}^{2}$

graph{(3x-1)^2 [-1.717, 2.128, -0.713, 1.209]}