# How do you find the roots, real and imaginary, of y=-2(x +3)^2-5x + 6 using the quadratic formula?

Feb 16, 2016

Roots: $x \in \left\{- 7.723 , - 0.777\right\}$

#### Explanation:

First convert the given equation:
$y = - 2 {\left(x + 3\right)}^{2} - 5 x + 6$ into standard quadratic form:

$y = - 2 \left({x}^{2} + 6 x + 9\right) - 5 x + 6$

$y = - 2 {x}^{2} - 12 x - 18 - 5 x + 6$

$y = - 2 {x}^{2} - 17 x - 12$

$\textcolor{w h i t e}{\text{XXX}} x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
with $a = - 2$, $b = - 17$, and $c = - 12$
x=(17+-sqrt(17^2-4(2)(12)))/(2(-2)
$\textcolor{w h i t e}{\text{XXX}} x \approx - 7.723$ or $x \approx - 0.777$