Question
$$\frac{1}{x}+\frac{1}{2}x = 0$$
Answer
$$ \begin{matrix}x_1 = \sqrt{ 2 } i & x_2 = -\sqrt{ 2 } i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x}+\frac{1}{2}x &= 0&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2\cdot\frac{1}{x}+x\cdot2\frac{1}{2}x &= x\cdot2\cdot0&& \text{cancel out the denominators} \\[1 em]2+x^2 &= 0&& \text{simplify left side} \\[1 em]x^2+2 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}+2 = 0 $ is a quadratic equation.
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