$$x-\frac{1}{x} = \frac{1}{2}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 1 }{ 4 }-\dfrac{\sqrt{ 17 }}{ 4 } & x_2 = \dfrac{ 1 }{ 4 }+\dfrac{\sqrt{ 17 }}{ 4 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x-\frac{1}{x} &= \frac{1}{2}&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2x-x\cdot2\cdot\frac{1}{x} &= x\cdot2\cdot\frac{1}{2}&& \text{cancel out the denominators} \\[1 em]2x^2-2 &= x&& \text{move all terms to the left hand side } \\[1 em]2x^2-2-x &= 0&& \text{simplify left side} \\[1 em]2x^2-x-2 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}-x-2 = 0 $ is a quadratic equation.
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