x^2=1/(x+1)

asked by guest
on Nov 21, 2024 at 9:52 pm



You asked:

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

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= - \frac{1}{3} + \sqrt[3]{\frac{25}{54} + \frac{\sqrt{69}}{18}} + \frac{1}{9 \sqrt[3]{\frac{25}{54} + \frac{\sqrt{69}}{18}}} \approx 0.75487767\\x &= - \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}}{2} - \frac{1}{3} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}}{2}\right) \approx -0.87743883 + 0.74486177 i\\x &= - \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}}{2} - \frac{1}{3} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{25}{54}}}\right) \approx -0.87743883 -0.74486177 i\end{aligned}\]

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

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