5x^4-12x^3-6x=0

solve for x

asked by guest
on Sep 22, 2024 at 1:36 pm



You asked:

Solve the equation \[5 \cdot {x}^{4} - 12 \cdot {x}^{3} - 6 x = 0\] for the unknown \(x\).

MathBot Answer:

The 4 solutions to the equation are: \[\begin{aligned}x &= 0\\x &= \frac{4}{5} + \sqrt[3]{\frac{139}{125} + \frac{\sqrt{609}}{25}} + \frac{16}{25 \sqrt[3]{\frac{139}{125} + \frac{\sqrt{609}}{25}}} \approx 2.5802436\\x &= - \frac{\sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}}{2} - \frac{8}{25 \sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}} + \frac{4}{5} + i \left(- \frac{8 \sqrt{3}}{25 \sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}}{2}\right) \approx -0.090121795 + 0.67598109 i\\x &= - \frac{\sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}}{2} - \frac{8}{25 \sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}} + \frac{4}{5} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}}{2} + \frac{8 \sqrt{3}}{25 \sqrt[3]{\frac{\sqrt{609}}{25} + \frac{139}{125}}}\right) \approx -0.090121795 -0.67598109 i\end{aligned}\]

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

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