$$n-\frac{5}{n}+8 = \frac{2}{7}$$
Answer
$$ \begin{matrix}n_1 = -\dfrac{ 27 }{ 7 }-\dfrac{\sqrt{ 974 }}{ 7 } & n_2 = -\dfrac{ 27 }{ 7 }+\dfrac{\sqrt{ 974 }}{ 7 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} n-\frac{5}{n}+8 &= \frac{2}{7}&& \text{multiply ALL terms by } \color{blue}{ n\cdot7 }. \\[1 em]n\cdot7n-n\cdot7\cdot\frac{5}{n}+n\cdot7\cdot8 &= n\cdot7\cdot\frac{2}{7}&& \text{cancel out the denominators} \\[1 em]7n^2-35+56n &= 2n&& \text{simplify left side} \\[1 em]7n^2+56n-35 &= 2n&& \text{move all terms to the left hand side } \\[1 em]7n^2+56n-35-2n &= 0&& \text{simplify left side} \\[1 em]7n^2+54n-35 &= 0&& \\[1 em] \end{aligned} $$
$ 7x^{2}+54x-35 = 0 $ is a quadratic equation.
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