Question
$$\frac{1}{x-1}+\frac{1}{x}+2 = \frac{5}{4}$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 5 }{ 6 }-\dfrac{\sqrt{ 73 }}{ 6 } & x_2 = -\dfrac{ 5 }{ 6 }+\dfrac{\sqrt{ 73 }}{ 6 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x-1}+\frac{1}{x}+2 &= \frac{5}{4}&& \text{multiply ALL terms by } \color{blue}{ (x-1)x\cdot4 }. \\[1 em](x-1)x\cdot4\cdot\frac{1}{x-1}+(x-1)x\cdot4\cdot\frac{1}{x}+(x-1)x\cdot4\cdot2 &= (x-1)x\cdot4\cdot\frac{5}{4}&& \text{cancel out the denominators} \\[1 em]4x+4x-4+8x^2-8x &= 5x^2-5x&& \text{simplify left side} \\[1 em]8x-4+8x^2-8x &= 5x^2-5x&& \\[1 em]8x^2-4 &= 5x^2-5x&& \text{move all terms to the left hand side } \\[1 em]8x^2-4-5x^2+5x &= 0&& \text{simplify left side} \\[1 em]3x^2+5x-4 &= 0&& \\[1 em] \end{aligned} $$
$ 3x^{2}+5x-4 = 0 $ is a quadratic equation.
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