$$\frac{x+1}{7}-\frac{1}{x} = \frac{8}{5}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 51 }{ 10 }-\dfrac{\sqrt{ 3301 }}{ 10 } & x_2 = \dfrac{ 51 }{ 10 }+\dfrac{\sqrt{ 3301 }}{ 10 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x+1}{7}-\frac{1}{x} &= \frac{8}{5}&& \text{multiply ALL terms by } \color{blue}{ 7x\cdot5 }. \\[1 em]7x\cdot5 \cdot \frac{x+1}{7}-7x\cdot5\cdot\frac{1}{x} &= 7x\cdot5\cdot\frac{8}{5}&& \text{cancel out the denominators} \\[1 em]5x^2+5x-35 &= 56x&& \text{move all terms to the left hand side } \\[1 em]5x^2+5x-35-56x &= 0&& \text{simplify left side} \\[1 em]5x^2-51x-35 &= 0&& \\[1 em] \end{aligned} $$
$ 5x^{2}-51x-35 = 0 $ is a quadratic equation.
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