$$x-\frac{1}{x} = 1$$
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
$$ \begin{aligned} x-\frac{1}{x} &= 1&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]xx-x\cdot\frac{1}{x} &= x\cdot1&& \text{cancel out the denominators} \\[1 em]x^2-1 &= x&& \text{move all terms to the left hand side } \\[1 em]x^2-1-x &= 0&& \text{simplify left side} \\[1 em]x^2-x-1 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}-x-1 = 0 $ is a quadratic equation.
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