$$\frac{7}{x}-\frac{1}{3}x = \frac{8}{3}$$
Answer
$$ \begin{matrix}x_1 = -4-\sqrt{ 37 } & x_2 = -4+\sqrt{ 37 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{7}{x}-\frac{1}{3}x &= \frac{8}{3}&& \text{multiply ALL terms by } \color{blue}{ x\cdot3 }. \\[1 em]x\cdot3\cdot\frac{7}{x}-x\cdot3\frac{1}{3}x &= x\cdot3\cdot\frac{8}{3}&& \text{cancel out the denominators} \\[1 em]21-x^2 &= 8x&& \text{simplify left side} \\[1 em]-x^2+21 &= 8x&& \text{move all terms to the left hand side } \\[1 em]-x^2+21-8x &= 0&& \text{simplify left side} \\[1 em]-x^2-8x+21 &= 0&& \\[1 em] \end{aligned} $$
$ -x^{2}-8x+21 = 0 $ is a quadratic equation.
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