[(1/x^2-5x)] = [(x+7)/x] -1

asked by guest
on Jan 19, 2025 at 2:21 am



You asked:

Solve the equation \(\frac{1}{{x}^{2}} - 5 x = \frac{x + 7}{x} - 1\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}} - \frac{7}{15 \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}} \approx 0.14086077\\x &= - \frac{\sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}}{2} + \frac{7}{30 \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}} + i \left(\frac{7 \sqrt{3}}{30 \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}}{2}\right) \approx -0.070430384 + 1.1894878 i\\x &= - \frac{\sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}}{2} + \frac{7}{30 \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}}{2} - \frac{7 \sqrt{3}}{30 \sqrt[3]{\frac{1}{10} + \frac{\sqrt{22605}}{450}}}\right) \approx -0.070430384 -1.1894878 i\end{aligned}\]

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

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