# How do you find the roots, real and imaginary, of y=-9x^2 -22x-55  using the quadratic formula?

May 24, 2017

$x = - \frac{11 + i \sqrt{384}}{9} , - \frac{11 - i \sqrt{384}}{9}$

Refer to the process in the explanation.

#### Explanation:

Substitute $0$ for $y$. The roots are the values for $x$ when $y$ equals $0$. Since the graph is a parabola, there will be two roots.

$- 9 {x}^{2} - 22 x - 55 = 0$ is a quadratic equation in standard form:

$a {x}^{2} + b x + c$, where $a = - 9$, $b = - 22$, $c = - 55$.

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Insert the values for $a , b , \mathmr{and} c$ into the formula.

$x = \frac{- \left(- 22\right) + \sqrt{{\left(- 22\right)}^{2} - 4 \cdot - 9 \cdot - 55}}{2 \cdot - 9}$

Simplify.

$x = \frac{22 \pm \sqrt{484 - 1980}}{-} 18$

Simplify.

$x = \frac{22 \pm \sqrt{- 1496}}{-} 18$

Prime factorize $- 1496$

$x = \frac{22 \pm \sqrt{- 1 \times 2 \times 2 \times 2 \times 11 \times 17}}{-} 18$
http://www.calculatorsoup.com/calculators/math/prime-factors.php

Simplify.

$x = \frac{22 \pm i \sqrt{- 1 \times {2}^{2} \times 2 \times 11 \times 17}}{-} 18$

Simplify.

$x = \frac{22 \pm 2 i \sqrt{384}}{-} 18$

Simplify. (Divide $22$, $2$, $18$ by $2$.)

$x = - \frac{11 + i \sqrt{384}}{9} ,$$x = - \frac{11 - i \sqrt{384}}{9}$