Math Problem Solver

The solutions are given by the formula: \[x = \left(-1 + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \cos{\left(\frac{2 \pi k}{7} \right)}}{37}\right) + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \sin{\left(\frac{2 \pi k}{7} \right)}}{37} i\] where \(k\) is an integer in \(0 \le k < 7\).

There are \(6\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= -1 + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50}}{37} \\&\approx 0.043953568\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= -1 + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \cos{\left(\frac{2 \pi}{7} \right)}}{37} + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} i \sin{\left(\frac{2 \pi}{7} \right)}}{37} \\&\approx -0.3491056 + 0.81619577 i\\x &= -1 - \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \cos{\left(\frac{3 \pi}{7} \right)}}{37} + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} i \sin{\left(\frac{3 \pi}{7} \right)}}{37} \\&\approx -1.2323015 + 1.0177795 i\\x &= -1 - \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \cos{\left(\frac{\pi}{7} \right)}}{37} + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} i \sin{\left(\frac{\pi}{7} \right)}}{37} \\&\approx -1.9405697 + 0.45295448 i\\x &= -1 - \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \cos{\left(\frac{\pi}{7} \right)}}{37} - \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} i \sin{\left(\frac{\pi}{7} \right)}}{37} \\&\approx -1.9405697 -0.45295448 i\\x &= -1 - \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \cos{\left(\frac{3 \pi}{7} \right)}}{37} - \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} i \sin{\left(\frac{3 \pi}{7} \right)}}{37} \\&\approx -1.2323015 -1.0177795 i\\x &= -1 + \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} \cos{\left(\frac{2 \pi}{7} \right)}}{37} - \frac{37^{\frac{6}{7}} \cdot \sqrt[7]{50} i \sin{\left(\frac{2 \pi}{7} \right)}}{37} \\&\approx -0.3491056 -0.81619577 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).