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