Question
$$\frac{1}{6}x+\frac{9}{x} = \frac{14}{3}$$
Answer
$$ \begin{matrix}x_1 = 14-\sqrt{ 142 } & x_2 = 14+\sqrt{ 142 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{6}x+\frac{9}{x} &= \frac{14}{3}&& \text{multiply ALL terms by } \color{blue}{ 6x\cdot3 }. \\[1 em]6x\cdot3 \cdot \frac{1}{6}x+6x\cdot3\cdot\frac{9}{x} &= 6x\cdot3\cdot\frac{14}{3}&& \text{cancel out the denominators} \\[1 em]3x^2+162 &= 84x&& \text{move all terms to the left hand side } \\[1 em]3x^2+162-84x &= 0&& \text{simplify left side} \\[1 em]3x^2-84x+162 &= 0&& \\[1 em] \end{aligned} $$
$ 3x^{2}-84x+162 = 0 $ is a quadratic equation.
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