$$\frac{1}{x}+\frac{1}{2x} = \frac{2}{3}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 2 }{ 3 }+\dfrac{\sqrt{ 14 }}{ 3 }i & x_2 = \dfrac{ 2 }{ 3 }-\dfrac{\sqrt{ 14 }}{ 3 }i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x}+\frac{1}{2x} &= \frac{2}{3}&& \text{multiply ALL terms by } \color{blue}{ x\cdot2\cdot3 }. \\[1 em]x\cdot2\cdot3\cdot\frac{1}{x}+x\cdot2\cdot3\cdot\frac{1}{2x} &= x\cdot2\cdot3\cdot\frac{2}{3}&& \text{cancel out the denominators} \\[1 em]6+3x^2 &= 4x&& \text{simplify left side} \\[1 em]3x^2+6 &= 4x&& \text{move all terms to the left hand side } \\[1 em]3x^2+6-4x &= 0&& \text{simplify left side} \\[1 em]3x^2-4x+6 &= 0&& \\[1 em] \end{aligned} $$
$ 3x^{2}-4x+6 = 0 $ is a quadratic equation.
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