# How do you solve x^2- x + 41 using the quadratic formula?

Jul 19, 2018

$x = \frac{1 + i \sqrt{163}}{2}$ and $x = \frac{1 - i \sqrt{163}}{2}$

#### Explanation:

We can find the roots of any quadratic of the form $a {x}^{2} + b x + c$ with the Quadratic Formula

$\overline{\underline{|} \textcolor{w h i t e}{\frac{2}{2}} x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a} \textcolor{w h i t e}{\frac{2}{2}} |}$

We have the quadratic ${x}^{2} - x + 41$, where $a = 1 , b = - 1$ and $c = 41$. Plugging these values in, we get

$x = \frac{1 \pm \sqrt{1 - 4 \left(1 \cdot 41\right)}}{2}$

This simplifies to

$x = \frac{1 \pm \sqrt{- 163}}{2}$ or

$x = \frac{1}{2} \pm \frac{\sqrt{- 163}}{2}$

We can rewrite $\sqrt{- 163}$ as $\sqrt{163} \cdot \sqrt{- 1}$, or $i \sqrt{3}$. Doing this, we now have

$x = \frac{1 + i \sqrt{163}}{2}$ and $x = \frac{1 - i \sqrt{163}}{2}$

Hope this helps!