$a^{3}$ -a=2

asked by guest
on Oct 24, 2024 at 10:25 am



You asked:

Solve the equation \({a}^{3} - a = 2\) for the variable \(a\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}a &= \sqrt[3]{1 + \frac{\sqrt{78}}{9}} + \frac{1}{3 \sqrt[3]{1 + \frac{\sqrt{78}}{9}}} \approx 1.5213797\\a &= - \frac{\sqrt[3]{\frac{\sqrt{78}}{9} + 1}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{78}}{9} + 1}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{78}}{9} + 1}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{78}}{9} + 1}}{2}\right) \approx -0.76068985 + 0.85787363 i\\a &= - \frac{\sqrt[3]{\frac{\sqrt{78}}{9} + 1}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{78}}{9} + 1}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{78}}{9} + 1}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{78}}{9} + 1}}\right) \approx -0.76068985 -0.85787363 i\end{aligned}\]

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

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