$X^{3}$ =30.X
You asked:
Solve the equation \({X}^{3} = 30\) for the variable \(X\).
MathBot Answer:
The solutions are given by the formula: \[X = \sqrt[3]{30} \cos{\left(\frac{2 \pi k}{3} \right)} + \sqrt[3]{30} \sin{\left(\frac{2 \pi k}{3} \right)} i\] 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: \[\begin{aligned}X &= \sqrt[3]{30} \\&\approx 3.1072325\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}X &= - \frac{\sqrt[3]{30}}{2} + \frac{\sqrt[3]{10} \cdot 3^{\frac{5}{6}} i}{2} \\&\approx -1.5536163 + 2.6909423 i\\X &= - \frac{\sqrt[3]{30}}{2} - \frac{\sqrt[3]{10} \cdot 3^{\frac{5}{6}} i}{2} \\&\approx -1.5536163 -2.6909423 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).