Question
$$\frac{4}{x} = \frac{8}{5}x+4$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 5 }{ 4 }-\dfrac{\sqrt{ 65 }}{ 4 } & x_2 = -\dfrac{ 5 }{ 4 }+\dfrac{\sqrt{ 65 }}{ 4 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{4}{x} &= \frac{8}{5}x+4&& \text{multiply ALL terms by } \color{blue}{ x\cdot5 }. \\[1 em]x\cdot5\cdot\frac{4}{x} &= x\cdot5 \cdot \frac{8}{5}x+x\cdot5\cdot4&& \text{cancel out the denominators} \\[1 em]20 &= 8x^2+20x&& \text{move all terms to the left hand side } \\[1 em]20-8x^2-20x &= 0&& \text{simplify left side} \\[1 em]-8x^2-20x+20 &= 0&& \\[1 em] \end{aligned} $$
$ -8x^{2}-20x+20 = 0 $ is a quadratic equation.
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