Question
$$x+\frac{1}{x} = \frac{10}{29}$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 5 }{ 29 }+\dfrac{ 4 \sqrt{ 51}}{ 29 }i & x_2 = \dfrac{ 5 }{ 29 }-\dfrac{ 4 \sqrt{ 51}}{ 29 }i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x+\frac{1}{x} &= \frac{10}{29}&& \text{multiply ALL terms by } \color{blue}{ x\cdot29 }. \\[1 em]x\cdot29x+x\cdot29\cdot\frac{1}{x} &= x\cdot29\cdot\frac{10}{29}&& \text{cancel out the denominators} \\[1 em]29x^2+29 &= 10x&& \text{move all terms to the left hand side } \\[1 em]29x^2+29-10x &= 0&& \text{simplify left side} \\[1 em]29x^2-10x+29 &= 0&& \\[1 em] \end{aligned} $$
$ 29x^{2}-10x+29 = 0 $ is a quadratic equation.
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