(-14/3) + (x^2)/2 + (5/24)*x^3 = 0

asked by guest
on Jan 15, 2025 at 10:56 am



You asked:

Solve the equation \(\frac{-14}{3} + \frac{{x}^{2}}{2} + \frac{5}{24} {x}^{3} = 0\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= - \frac{4}{5} + 2 \sqrt[3]{\frac{167}{125} + \frac{\sqrt{1113}}{25}} + \frac{8}{25 \sqrt[3]{\frac{167}{125} + \frac{\sqrt{1113}}{25}}} \approx 2.2054121\\x &= - \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}} - \frac{4}{5} - \frac{4}{25 \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}}} + i \left(- \frac{4 \sqrt{3}}{25 \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}}} + \sqrt{3} \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}}\right) \approx -2.3027061 + 2.203265 i\\x &= - \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}} - \frac{4}{5} - \frac{4}{25 \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}}} + i \left(- \sqrt{3} \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}} + \frac{4 \sqrt{3}}{25 \sqrt[3]{\frac{\sqrt{1113}}{25} + \frac{167}{125}}}\right) \approx -2.3027061 -2.203265 i\end{aligned}\]

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

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