$x^{3}=-1$
You asked:
Solve the equation \({x}^{3} = -1\) for the variable \(x\).
MathBot Answer:
The solutions are given by the formula: \[x = \cos{\left(\frac{\pi}{3} + \frac{2 \pi k}{3} \right)} + \sin{\left(\frac{\pi}{3} + \frac{2 \pi k}{3} \right)} i\] 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: \[\begin{aligned}x &= -1\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{1}{2} + \frac{\sqrt{3} i}{2} \\&\approx 0.5 + 0.8660254 i\\x &= \frac{1}{2} - \frac{\sqrt{3} i}{2} \\&\approx 0.5 -0.8660254 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).