# How do you find the roots, real and imaginary, of y= 12x^2 - 4x + 8- (-x-1)^2  using the quadratic formula?

Oct 23, 2017

Roots

$x = \frac{3 + 2 i \sqrt{17}}{11} ,$$\frac{3 - 2 i \sqrt{17}}{11}$

#### Explanation:

Solve:

$y = 12 {x}^{2} - 4 x + 8 - {\left(- x - 1\right)}^{2}$

First simplify the parentheses.

$y = 12 {x}^{2} - 4 x + 8 - \left({x}^{2} + 2 x + 1\right)$

Simplify.

$y = 12 {x}^{2} - 4 x + 8 - {x}^{2} - 2 x - 1$

Gather like terms.

$y = \left(12 {x}^{2} - {x}^{2}\right) + \left(- 4 x - 2 x\right) + \left(8 - 1\right)$

Combine like terms.

$y = 11 {x}^{2} - 6 x + 7$ is a quadratic equation in standard form:

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

where:

$a = 11$, $b = - 6$, and $c = 7$

Substitute $0$ for $y$.

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

Plug the known values into the formula.

$x = \frac{- \left(- 6\right) \pm \sqrt{{\left(- 6\right)}^{2} - 4 \cdot 11 \cdot 7}}{2 \cdot 11}$

Simplify.

$x = \frac{6 \pm \sqrt{36 - 308}}{22}$

Simplify.

$x = \frac{6 \pm \sqrt{- 272}}{22}$

Prime factorize $272$.

$x = \frac{6 \pm \left(- \left(2 \times 2\right) \times \left(2 \times 2\right) \times 17\right)}{22}$

Simplify.

$x = \frac{6 \pm 4 i \sqrt{17}}{22}$

Reduce.

$x = \frac{\frac{6 \pm 4 i \sqrt{17}}{22}}{2}$

$x = \frac{3 \pm 2 i \sqrt{17}}{11}$

Roots

$x = \frac{3 + 2 i \sqrt{17}}{11} ,$$\frac{3 - 2 i \sqrt{17}}{11}$