$$x+\frac{1}{x} = 5$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 5 }{ 2 }-\dfrac{\sqrt{ 21 }}{ 2 } & x_2 = \dfrac{ 5 }{ 2 }+\dfrac{\sqrt{ 21 }}{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x+\frac{1}{x} &= 5&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]xx+x\cdot\frac{1}{x} &= x\cdot5&& \text{cancel out the denominators} \\[1 em]x^2+1 &= 5x&& \text{move all terms to the left hand side } \\[1 em]x^2+1-5x &= 0&& \text{simplify left side} \\[1 em]x^2-5x+1 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}-5x+1 = 0 $ is a quadratic equation.
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