a^3 + 12a + 117 = 0

asked by guest
on Nov 18, 2024 at 5:25 am



You asked:

Solve the equation \({a}^{3} + 12 a + 117 = 0\) for the variable \(a\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}a &= \frac{12}{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}} - \frac{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}}{3} \approx -4.08198\\a &= - \frac{6}{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}} + \frac{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}}{6} + i \left(\frac{6 \sqrt{3}}{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}}{6}\right) \approx 2.04099 + 4.9494364 i\\a &= - \frac{6}{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}} + \frac{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}}{6} - \frac{6 \sqrt{3}}{\sqrt[3]{\frac{3159}{2} + \frac{27 \sqrt{13945}}{2}}}\right) \approx 2.04099 -4.9494364 i\end{aligned}\]

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

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