6k^4 - 14k^3 + 10k^2 - 1 = 0

asked by guest
on Feb 13, 2025 at 8:15 pm



You asked:

Solve the equation \(6 {k}^{4} - 14 {k}^{3} + 10 {k}^{2} - 1 = 0\) for the variable \(k\).

MathBot Answer:

The 4 solutions to the equation are: \[\begin{aligned}k &= \frac{7}{12} - \frac{\sqrt{\frac{1}{4} + 2 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}} + \frac{7}{162 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}}}}}{2} - \frac{\sqrt{\frac{1}{2} - 2 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}} - \frac{7}{162 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}}} + \frac{77}{108 \sqrt{\frac{1}{4} + 2 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}} + \frac{7}{162 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}}}}}}}{2} \approx -0.26587486\\k &= \frac{7}{12} + \frac{\sqrt{\frac{1}{2} - 2 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}} - \frac{7}{162 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}}} + \frac{77}{108 \sqrt{\frac{1}{4} + 2 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}} + \frac{7}{162 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}}}}}}}{2} - \frac{\sqrt{\frac{1}{4} + 2 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}} + \frac{7}{162 \sqrt[3]{\frac{257}{23328} + \frac{\sqrt{6729}}{7776}}}}}{2} \approx 0.45175895\\k &= \frac{\sqrt{\frac{7}{162 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}} + \frac{1}{4} + 2 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}}}{2} + \frac{7}{12} - \frac{\sqrt{- \frac{77}{108 \sqrt{\frac{7}{162 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}} + \frac{1}{4} + 2 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}}} - 2 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}} - \frac{7}{162 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}} + \frac{1}{2}}}{2} \approx 1.0737246 -0.4844758 i\\k &= \frac{\sqrt{\frac{7}{162 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}} + \frac{1}{4} + 2 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}}}{2} + \frac{7}{12} + \frac{\sqrt{- \frac{77}{108 \sqrt{\frac{7}{162 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}} + \frac{1}{4} + 2 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}}} - 2 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}} - \frac{7}{162 \sqrt[3]{\frac{\sqrt{6729}}{7776} + \frac{257}{23328}}} + \frac{1}{2}}}{2} \approx 1.0737246 + 0.4844758 i\end{aligned}\]

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