$$x = \frac{6}{x}-1$$
Answer
$$ \begin{matrix}x_1 = 2 & x_2 = -3 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x &= \frac{6}{x}-1&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]xx &= x\cdot\frac{6}{x}-x\cdot1&& \text{cancel out the denominators} \\[1 em]x^2 &= 6-x&& \text{move all terms to the left hand side } \\[1 em]x^2-6+x &= 0&& \text{simplify left side} \\[1 em]x^2+x-6 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}+x-6 = 0 $ is a quadratic equation.
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