# How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given 4x^2+7=9x?

Nov 21, 2016

#### Answer:

$\Delta = - 31 < 0 \text{ }$ so roots are complex.

$x = \frac{9}{8} \pm i \frac{\sqrt{31}}{8}$

#### Explanation:

1) rearrange to the form ""ax^2+bx+c=0

$4 {x}^{2} + 7 = 9 x$

$4 {x}^{2} - 9 x + 7 = 0$

2) check the discriminant$\text{ } \Delta = {b}^{2} - 4 a c$

$\Delta = {\left(- 9\right)}^{2} - 4 \times 4 \times 7$

$\Delta = 81 - 112 = - 31$

types of roots

$\Delta > 0 \text{ }$real distinct roots

$\Delta = 0 \text{ }$real and equal roots

$\Delta < 0 \text{ }$roots are complex.

In this case $\text{ "Delta=-31<0" }$ so roots are complex.

3) so the exact solutions

using the formula

$x = \frac{- b \pm \sqrt{\Delta}}{2 a}$

$x = \frac{- \left(- 9\right) \pm \sqrt{- 31}}{2 \times 4}$

$x = \frac{9 \pm i \sqrt{31}}{8}$

$x = \frac{9}{8} \pm i \frac{\sqrt{31}}{8}$