# How do you find the roots, real and imaginary, of y=-2x^2+6x-(2x-7)^2  using the quadratic formula?

Jan 29, 2018

$x = \frac{17 + i \sqrt{- 5}}{6} , \frac{17 - i \sqrt{- 5}}{6}$

#### Explanation:

First, expand the equation.

$- 2 {x}^{2} + 6 x - \left(4 {x}^{2} - 28 x + 49\right)$

$- 2 {x}^{2} - 4 {x}^{2} + 6 x + 28 x - 49$

$- 6 {x}^{2} + 34 x - 49$

$6 {x}^{2} - 34 + 49$

The quadratic equation is $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$.

Here, $a = 6 , b = - 34 , c = 49$

Input:

$\frac{- \left(- 34\right) \pm \sqrt{{\left(- 34\right)}^{2} - 4 \cdot 6 \cdot 49}}{2 \cdot 6}$

$\frac{34 \pm \sqrt{1156 - 1176}}{12}$

$\frac{34 \pm \sqrt{- 20}}{12}$

$\frac{34 + \sqrt{- 20}}{12} , \frac{34 - \sqrt{- 20}}{12}$

$\frac{34 + 2 i \sqrt{- 5}}{12} , \frac{34 - 2 i \sqrt{- 5}}{12}$

$x = \frac{17 + i \sqrt{- 5}}{6} , \frac{17 - i \sqrt{- 5}}{6}$