Question
$$\frac{1}{x} = \frac{7}{6x}+1$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 3 }{ 7 }-\dfrac{\sqrt{ 51 }}{ 7 } & x_2 = -\dfrac{ 3 }{ 7 }+\dfrac{\sqrt{ 51 }}{ 7 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x} &= \frac{7}{6x}+1&& \text{multiply ALL terms by } \color{blue}{ x\cdot6 }. \\[1 em]x\cdot6\cdot\frac{1}{x} &= x\cdot6\cdot\frac{7}{6x}+x\cdot6\cdot1&& \text{cancel out the denominators} \\[1 em]6 &= 7x^2+6x&& \text{move all terms to the left hand side } \\[1 em]6-7x^2-6x &= 0&& \text{simplify left side} \\[1 em]-7x^2-6x+6 &= 0&& \\[1 em] \end{aligned} $$
$ -7x^{2}-6x+6 = 0 $ is a quadratic equation.
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