$$\frac{3}{4}x+\frac{2}{x}+6 = \frac{5}{8}$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 43 }{ 12 }-\dfrac{\sqrt{ 1465 }}{ 12 } & x_2 = -\dfrac{ 43 }{ 12 }+\dfrac{\sqrt{ 1465 }}{ 12 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{3}{4}x+\frac{2}{x}+6 &= \frac{5}{8}&& \text{multiply ALL terms by } \color{blue}{ 4x\cdot8 }. \\[1 em]4x\cdot8 \cdot \frac{3}{4}x+4x\cdot8\cdot\frac{2}{x}+4x\cdot8\cdot6 &= 4x\cdot8\cdot\frac{5}{8}&& \text{cancel out the denominators} \\[1 em]24x^2+64+192x &= 20x&& \text{simplify left side} \\[1 em]24x^2+192x+64 &= 20x&& \text{move all terms to the left hand side } \\[1 em]24x^2+192x+64-20x &= 0&& \text{simplify left side} \\[1 em]24x^2+172x+64 &= 0&& \\[1 em] \end{aligned} $$
$ 24x^{2}+172x+64 = 0 $ is a quadratic equation.
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