Math Problem Solver

The solutions are given by the formula: $$x = \frac{2^{\frac{9}{10}} \cos{\left(\frac{\pi k}{5} \right)}}{2} + \frac{2^{\frac{9}{10}} \sin{\left(\frac{\pi k}{5} \right)}}{2} i$$ where $k$ is an integer in $0 \le k < 10$.

There are $2$ real solutions.

There are $8$ solutions with nonzero imaginary part.

Real solutions: $$\begin{aligned}x &= \frac{2^{\frac{9}{10}}}{2} \\&\approx 0.93303299\\x &= - \frac{2^{\frac{9}{10}}}{2} \\&\approx -0.93303299\end{aligned}$$

Solutions with nonzero imaginary part: $$\begin{aligned}x &= \frac{2^{\frac{9}{10}} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} + \frac{2^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} \\&\approx 0.75483955 + 0.54842303 i\\x &= \frac{2^{\frac{9}{10}} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} + \frac{2^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} \\&\approx 0.28832305 + 0.88736711 i\\x &= \frac{2^{\frac{9}{10}} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} + \frac{2^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} \\&\approx -0.28832305 + 0.88736711 i\\x &= \frac{2^{\frac{9}{10}} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{2} + \frac{2^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} \\&\approx -0.75483955 + 0.54842303 i\\x &= \frac{2^{\frac{9}{10}} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right)}{2} - \frac{2^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} \\&\approx -0.75483955 -0.54842303 i\\x &= \frac{2^{\frac{9}{10}} \cdot \left(\frac{1}{4} - \frac{\sqrt{5}}{4}\right)}{2} - \frac{2^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} \\&\approx -0.28832305 -0.88736711 i\\x &= \frac{2^{\frac{9}{10}} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} - \frac{2^{\frac{9}{10}} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{2} \\&\approx 0.28832305 -0.88736711 i\\x &= \frac{2^{\frac{9}{10}} \cdot \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{2} - \frac{2^{\frac{9}{10}} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} \\&\approx 0.75483955 -0.54842303 i\end{aligned}$$

$i$ is the imaginary unit, defined as $i^2 = -1$.