x^3 = -2
You asked:
Solve the equation \({x}^{3} = -2\) for the variable \(x\).
MathBot Answer:
The solutions are given by the formula: \[x = \sqrt[3]{2} \cos{\left(\frac{\pi}{3} + \frac{2 \pi k}{3} \right)} + \sqrt[3]{2} \sin{\left(\frac{\pi}{3} + \frac{2 \pi k}{3} \right)} i\] where \(k\) is an integer in \(0 \le k < 3\).
There is \(1\) real solution.There are \(2\) solutions with nonzero imaginary part.
Real solutions: \[\begin{aligned}x &= - \sqrt[3]{2} \\&\approx -1.259921\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{\sqrt[3]{2}}{2} + \frac{\sqrt[3]{2} \sqrt{3} i}{2} \\&\approx 0.62996052 + 1.0911236 i\\x &= \frac{\sqrt[3]{2}}{2} - \frac{\sqrt[3]{2} \sqrt{3} i}{2} \\&\approx 0.62996052 -1.0911236 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).