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

Apr 18, 2017

Roots are $\left(5 + i \sqrt{71}\right) 4$ and $\left(5 - i \sqrt{71}\right) 4$

#### Explanation:

$y = - {x}^{2} - x - 3 - {\left(x - 3\right)}^{2}$

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

= $- {x}^{2} - x - 3 - {x}^{2} + 6 x - 9$

= $- 2 {x}^{2} + 5 x - 12$

Using quadratic formula, the roots are (-5+-sqrt(5^2-4×(-2)×(-12)))/(2×(-2)

= $\frac{- 5 \pm \sqrt{25 - 96}}{- 4}$

= $\frac{5 \pm \sqrt{- 71}}{4}$

i.e. roots are $\left(5 + i \sqrt{71}\right) 4$ and $\left(5 - i \sqrt{71}\right) 4$