$$\frac{1}{x}+\frac{1}{x+1} = \frac{7}{12}$$
Answer
$$ \begin{matrix}x_1 = 3 & x_2 = -\dfrac{ 4 }{ 7 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x}+\frac{1}{x+1} &= \frac{7}{12}&& \text{multiply ALL terms by } \color{blue}{ x(x+1)\cdot12 }. \\[1 em]x(x+1)\cdot12\cdot\frac{1}{x}+x(x+1)\cdot12\cdot\frac{1}{x+1} &= x(x+1)\cdot12\cdot\frac{7}{12}&& \text{cancel out the denominators} \\[1 em]12x+12+12x &= 7x^2+7x&& \text{simplify left side} \\[1 em]24x+12 &= 7x^2+7x&& \text{move all terms to the left hand side } \\[1 em]24x+12-7x^2-7x &= 0&& \text{simplify left side} \\[1 em]-7x^2+17x+12 &= 0&& \\[1 em] \end{aligned} $$
$ -7x^{2}+17x+12 = 0 $ is a quadratic equation.
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