$a^{3}$-a=-2
You asked:
Solve the equation \({a}^{3} - a = -2\) for the variable \(a\).
MathBot Answer:
The 3 solutions to the equation are: \[\begin{aligned}a &= - \frac{1}{\sqrt[3]{27 + 3 \sqrt{78}}} - \frac{\sqrt[3]{27 + 3 \sqrt{78}}}{3} \approx -1.5213797\\a &= \frac{1}{2 \sqrt[3]{3 \sqrt{78} + 27}} + \frac{\sqrt[3]{3 \sqrt{78} + 27}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{3 \sqrt{78} + 27}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{78} + 27}}{6}\right) \approx 0.76068985 + 0.85787363 i\\a &= \frac{1}{2 \sqrt[3]{3 \sqrt{78} + 27}} + \frac{\sqrt[3]{3 \sqrt{78} + 27}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{78} + 27}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{3 \sqrt{78} + 27}}\right) \approx 0.76068985 -0.85787363 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).