$$x+\frac{1}{x} = 2^{1/2}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 3 }}{ 2 }i & x_2 = \dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 3 }}{ 2 }i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x+\frac{1}{x} &= 2^{1/2}&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2x+x\cdot2\cdot\frac{1}{x} &= x\cdot2\cdot2^{1/2}&& \text{cancel out the denominators} \\[1 em]2x^2+2 &= 2x&& \text{move all terms to the left hand side } \\[1 em]2x^2+2-2x &= 0&& \text{simplify left side} \\[1 em]2x^2-2x+2 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}-2x+2 = 0 $ is a quadratic equation.
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