\frac{1}{x+1}+x=\frac{\left(x^{2}-x-1\right)}{x^{2}-1}\cdot\frac{2}{x-1}
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You asked:
Solve the equation
1 x + 1 + x = x 2 − x − 1 x 2 − 1 ⋅ 2 x − 1 \frac{1}{x + 1} + x = \frac{{x}^{2} - x - 1}{{x}^{2} - 1} \cdot \frac{2}{x - 1} x + 1 1 + x = x 2 − 1 x 2 − x − 1 ⋅ x − 1 2 for the variable
x x x .
MathBot Answer:
The complex solutions are: { 1 4 − i − 19 6 + 43 18 253 216 + 1722 i 72 3 − 1 4 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 + 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 , 1 4 − i − 19 6 + 43 18 253 216 + 1722 i 72 3 + 1 4 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 − 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 , 1 4 − 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 + i − 19 6 + 43 18 253 216 + 1722 i 72 3 + 1 4 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 , 1 4 + 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 + i − 19 6 + 43 18 253 216 + 1722 i 72 3 − 1 4 19 12 + 43 18 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 + 2 253 216 + 1722 i 72 3 2 } ∖ { 1 } \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\} ⎩ ⎨ ⎧ 4 1 − 2 i − 6 19 + 18 3 216 253 + 72 1722 i 43 − 4 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i 1 + 2 3 216 253 + 72 1722 i + 2 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i , 4 1 − 2 i − 6 19 + 18 3 216 253 + 72 1722 i 43 + 4 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i 1 + 2 3 216 253 + 72 1722 i − 2 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i , 4 1 − 2 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i + 2 i − 6 19 + 18 3 216 253 + 72 1722 i 43 + 4 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i 1 + 2 3 216 253 + 72 1722 i , 4 1 + 2 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i + 2 i − 6 19 + 18 3 216 253 + 72 1722 i 43 − 4 12 19 + 18 3 216 253 + 72 1722 i 43 + 2 3 216 253 + 72 1722 i 1 + 2 3 216 253 + 72 1722 i ⎭ ⎬ ⎫ ∖ { 1 }
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
