$$-1+\frac{1}{4}x = \frac{1}{x}$$
Answer
$$ \begin{matrix}x_1 = 2-2 \sqrt{ 2 } & x_2 = 2+2 \sqrt{ 2 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} -1+\frac{1}{4}x &= \frac{1}{x}&& \text{multiply ALL terms by } \color{blue}{ 4x }. \\[1 em]-4x\cdot1+4x \cdot \frac{1}{4}x &= 4x\cdot\frac{1}{x}&& \text{cancel out the denominators} \\[1 em]-4x+x^2 &= 4&& \text{simplify left side} \\[1 em]x^2-4x &= 4&& \text{move all terms to the left hand side } \\[1 em]x^2-4x-4 &= 0&& \\[1 em] \end{aligned} $$
$ x^{2}-4x-4 = 0 $ is a quadratic equation.
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