$$\frac{8}{z}-z = 4$$
Answer
$$ \begin{matrix}z_1 = -2-2 \sqrt{ 3 } & z_2 = -2+2 \sqrt{ 3 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{8}{z}-z &= 4&& \text{multiply ALL terms by } \color{blue}{ z }. \\[1 em]z\cdot\frac{8}{z}-zz &= z\cdot4&& \text{cancel out the denominators} \\[1 em]8-z^2 &= 4z&& \text{simplify left side} \\[1 em]-z^2+8 &= 4z&& \text{move all terms to the left hand side } \\[1 em]-z^2+8-4z &= 0&& \text{simplify left side} \\[1 em]-z^2-4z+8 &= 0&& \\[1 em] \end{aligned} $$
$ -x^{2}-4x+8 = 0 $ is a quadratic equation.
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