# How do you find the roots, real and imaginary, of y=-(x-3)^2-(2x-3)^2-10  using the quadratic formula?

Jan 23, 2018

$x = \frac{9 + i \sqrt{59}}{5} ,$$\frac{9 - i \sqrt{59}}{5}$

$x = \frac{9 + \sqrt{59} i}{5} ,$$\frac{9 - \sqrt{59} i}{5}$

#### Explanation:

Given:

$y = - {\left(x - 3\right)}^{2} - {\left(2 x - 3\right)}^{2} - 10$

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

$y = - \left({x}^{2} - 6 x + 9\right) - {\left(2 x - 3\right)}^{2} - 10$

Expand ${\left(2 x - 3\right)}^{2}$ to $4 {x}^{2} - 12 x + 9$.

$y = - \left({x}^{2} - 6 x + 9\right) - \left(4 {x}^{2} - 12 x + 9\right) - 10$

Distribute the $- 1$ through both expressions.

$y = - {x}^{2} + 6 x - 9 - 4 {x}^{2} + 12 x - 9 - 10$

Collect like terms.

$y = \left(- {x}^{2} - 4 {x}^{2}\right) + \left(6 x + 12 x\right) + \left(- 9 - 9 - 10\right)$

Simplify.

$y = - 5 {x}^{2} + 18 x - 28$ is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c$,

where:

$a = - 5$, $b = + 18$, and $c = - 28$

Find the roots using the quadratic formula.

The roots are the values for $x$ when $y = 0$, so first substitute $0$ for $y$.

$0 = - 5 {x}^{2} + 18 x - 28$

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

Plug in the known values.

$x = \frac{- 18 \pm \sqrt{{18}^{2} - 4 \cdot - 5 \cdot - 28}}{2 \cdot - 5}$

Simplify.

$x = \frac{- 18 \pm \sqrt{- 236}}{- 10}$

Prime factorize $- 236$.

$x = \frac{- 18 \pm \sqrt{- 1 \times 2 \times 2 \times 59}}{- 10}$

Simplify.

$x = \frac{- 18 \pm 2 i \sqrt{59}}{- 10}$

Two negatives make a positive.

$x = \frac{18 \pm 2 i \sqrt{59}}{10}$

Simplify by dividing by $2$.

$x = \frac{{\textcolor{red}{\cancel{\textcolor{b l a c k}{18}}}}^{9} \pm {\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}^{1} i \sqrt{59}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{10}}}} ^ 5$

$x = \frac{9 \pm i \sqrt{59}}{5}$

Solutions for $x$ (the roots)

$x = \frac{9 + i \sqrt{59}}{5} ,$$\frac{9 - i \sqrt{59}}{5}$

They can also be written with the imaginary number $i$ after the square root.

$x = \frac{9 + \sqrt{59} i}{5} ,$$\frac{9 - \sqrt{59} i}{5}$