$$\frac{1}{2}x+\frac{2}{x} = \frac{1}{x}$$
Answer
$$ \begin{matrix}x_1 = \sqrt{ 2 } i & x_2 = -\sqrt{ 2 } i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{2}x+\frac{2}{x} &= \frac{1}{x}&& \text{multiply ALL terms by } \color{blue}{ 2x }. \\[1 em]2x \cdot \frac{1}{2}x+2x\cdot\frac{2}{x} &= 2x\cdot\frac{1}{x}&& \text{cancel out the denominators} \\[1 em]x^2+4 &= 2&& \text{move all terms to the left hand side } \\[1 em]x^2+4-2 &= 0&& \text{simplify left side} \\[1 em]x^2+2 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}+2 = 0 $ is a quadratic equation.
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