\frac{1}{x+1}+x=\frac{\left(x^{2}-x-1\right)}{x^{2}-1}\cdot\frac{2}{x-1}

asked by guest
on Nov 23, 2024 at 1:27 pm



You asked:

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

MathBot Answer:

The complex solutions are: \[\left\{\frac{1}{4} - \frac{i \sqrt{- \frac{19}{6} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} - \frac{1}{4 \sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2} + \frac{\sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2}, \frac{1}{4} - \frac{i \sqrt{- \frac{19}{6} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + \frac{1}{4 \sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2} - \frac{\sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2}, \frac{1}{4} - \frac{\sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2} + \frac{i \sqrt{- \frac{19}{6} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + \frac{1}{4 \sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2}, \frac{1}{4} + \frac{\sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2} + \frac{i \sqrt{- \frac{19}{6} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} - \frac{1}{4 \sqrt{\frac{19}{12} + \frac{43}{18 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}} + 2 \sqrt[3]{\frac{253}{216} + \frac{\sqrt{1722} i}{72}}}}{2}\right\} \setminus \left\{1\right\}\]

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

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