The Algebra Gauntlet

Solve for $\textit{x}$ :

Now use $\textit{x}$ to find $\textit{a}$ , $\textit{b}$ , and $\textit{c}$ :

Set $\alpha$ and $\beta$ as shown below:

Solve for $\textit{k}$ :

Find the area of a regular hexagon with side length $\textit{k}$ . If the answer is in the form of $\frac { p\sqrt { q } }{ r }$ , give your answer as $2\cdot \frac { \sqrt { \frac { p }{ q } } }{ { (4{ r }^{ 2 }) }^{ \frac { 1 }{ 2 } } }$ , to the nearest tenths place.

Hints:

• $\textit{a}$ , $\textit{b}$ , and $\textit{c}$ are the side lengths of a right triangle.

• $1+{ e }^{ \pi i }=0$