Math Problem Solver

The solutions are given by the formula: $$x = \frac{5 \cdot 6^{\frac{2}{3}} \cos{\left(\frac{2 \pi k}{3} \right)}}{3} + \frac{5 \cdot 6^{\frac{2}{3}} \sin{\left(\frac{2 \pi k}{3} \right)}}{3} i$$ where $k$ is an integer in $0 \le k < 3$.

There are $2$ solutions with nonzero imaginary part.

Real solutions: $$\begin{aligned}x &= \frac{5 \cdot 6^{\frac{2}{3}}}{3} \\&\approx 5.5032121\end{aligned}$$

Solutions with nonzero imaginary part: $$\begin{aligned}x &= - \frac{5 \cdot 6^{\frac{2}{3}}}{6} + \frac{5 \cdot 2^{\frac{2}{3}} \cdot \sqrt[6]{3} i}{2} \\&\approx -2.751606 + 4.7659215 i\\x &= - \frac{5 \cdot 6^{\frac{2}{3}}}{6} - \frac{5 \cdot 2^{\frac{2}{3}} \cdot \sqrt[6]{3} i}{2} \\&\approx -2.751606 -4.7659215 i\end{aligned}$$

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