\(x^2+\frac{1}{x^2}+x-\frac{1}{x}-2=0\)
\(\Leftrightarrow x^2-2\cdot x\cdot\frac{1}{x}+\frac{1}{x^2}+x-\frac{1}{x}=0\)
\(\Leftrightarrow\left(x-\frac{1}{x}\right)^2+\left(x-\frac{1}{x}\right)=0\)
\(\Leftrightarrow\left(x-\frac{1}{x}\right)\left(x-\frac{1}{x}+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{1}{x}=0\\x-\frac{1}{x}+1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\frac{x^2-1}{x}=0\\\frac{x^2+x-1}{x}=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x^2-1=0\\x^2+x-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left(x-1\right)\left(x+1\right)=0\\x^2+2\cdot x\cdot\frac{1}{2}+\frac{1}{4}-\frac{5}{4}=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\pm1\\\left(x+\frac{1}{2}\right)^2=\frac{5}{4}=\left(\frac{\pm\sqrt{5}}{2}\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\pm1\\x=\frac{\pm\sqrt{5}-1}{2}\end{matrix}\right.\)
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