Question
$$\frac{1}{x-1}+\frac{1}{x+2} = \frac{5}{4}$$
Answer
$$ \begin{matrix}x_1 = 2 & x_2 = -\dfrac{ 7 }{ 5 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x-1}+\frac{1}{x+2} &= \frac{5}{4}&& \text{multiply ALL terms by } \color{blue}{ (x-1)(x+2)\cdot4 }. \\[1 em](x-1)(x+2)\cdot4\cdot\frac{1}{x-1}+(x-1)(x+2)\cdot4\cdot\frac{1}{x+2} &= (x-1)(x+2)\cdot4\cdot\frac{5}{4}&& \text{cancel out the denominators} \\[1 em]4x+8+4x-4 &= 5x^2+5x-10&& \text{simplify left side} \\[1 em]8x+4 &= 5x^2+5x-10&& \text{move all terms to the left hand side } \\[1 em]8x+4-5x^2-5x+10 &= 0&& \text{simplify left side} \\[1 em]-5x^2+3x+14 &= 0&& \\[1 em] \end{aligned} $$
$ -5x^{2}+3x+14 = 0 $ is a quadratic equation.
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