$$\frac{2}{x}-\frac{4}{3x} = \frac{1}{6}$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 16 }-\dfrac{\sqrt{ 385 }}{ 16 } & x_2 = -\dfrac{ 1 }{ 16 }+\dfrac{\sqrt{ 385 }}{ 16 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{2}{x}-\frac{4}{3x} &= \frac{1}{6}&& \text{multiply ALL terms by } \color{blue}{ x\cdot3\cdot6 }. \\[1 em]x\cdot3\cdot6\cdot\frac{2}{x}-x\cdot3\cdot6\cdot\frac{4}{3x} &= x\cdot3\cdot6\cdot\frac{1}{6}&& \text{cancel out the denominators} \\[1 em]36-24x^2 &= 3x&& \text{simplify left side} \\[1 em]-24x^2+36 &= 3x&& \text{move all terms to the left hand side } \\[1 em]-24x^2+36-3x &= 0&& \text{simplify left side} \\[1 em]-24x^2-3x+36 &= 0&& \\[1 em] \end{aligned} $$
$ -24x^{2}-3x+36 = 0 $ is a quadratic equation.
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