\(x^{2}+x-\frac{30}{x}-5=1\)

asked by guest
on Nov 25, 2024 at 7:30 pm



You asked:

Solve the equation \({x}^{2} + x - \frac{30}{x} - 5 = 1\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= - \frac{1}{3} + \sqrt[3]{\frac{377}{27} + \frac{\sqrt{1670}}{3}} + \frac{19}{9 \sqrt[3]{\frac{377}{27} + \frac{\sqrt{1670}}{3}}} \approx 3.3868678\\x &= - \frac{\sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}}{2} - \frac{19}{18 \sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}} - \frac{1}{3} + i \left(- \frac{19 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}}{2}\right) \approx -2.1934339 + 2.0116136 i\\x &= - \frac{\sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}}{2} - \frac{19}{18 \sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}} - \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}}{2} + \frac{19 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{1670}}{3} + \frac{377}{27}}}\right) \approx -2.1934339 -2.0116136 i\end{aligned}\]

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

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