2Q^3+2Q^2-2Q-501=0

asked by guest
on Oct 21, 2024 at 4:50 am



You asked:

Solve the equation \(2 \cdot {Q}^{3} + 2 \cdot {Q}^{2} - 2 Q - 501 = 0\) for the variable \(Q\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}Q &= - \frac{1}{3} + \sqrt[3]{\frac{13505}{108} + \frac{\sqrt{20264889}}{36}} + \frac{4}{9 \sqrt[3]{\frac{13505}{108} + \frac{\sqrt{20264889}}{36}}} \approx 6.0375891\\Q &= - \frac{\sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}}{2} - \frac{1}{3} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}} + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}}{2}\right) \approx -3.5187945 + 5.3951975 i\\Q &= - \frac{\sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}}{2} - \frac{1}{3} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{20264889}}{36} + \frac{13505}{108}}}\right) \approx -3.5187945 -5.3951975 i\end{aligned}\]

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

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