Question
$$\frac{1}{x}+\frac{2}{2x-3} = \frac{1}{x-2}$$
Answer
$$ \begin{matrix}x_1 = 1 & x_2 = 3 \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{1}{x}+\frac{2}{2x-3} &= \frac{1}{x-2}&& \text{multiply ALL terms by } \color{blue}{ x(2x-3)(x-2) }. \\[1 em]x(2x-3)(x-2)\cdot\frac{1}{x}+x(2x-3)(x-2)\cdot\frac{2}{2x-3} &= x(2x-3)(x-2)\cdot\frac{1}{x-2}&& \text{cancel out the denominators} \\[1 em]2x^2-7x+6+2x^2-4x &= 2x^2-3x&& \text{simplify left side} \\[1 em]4x^2-11x+6 &= 2x^2-3x&& \text{move all terms to the left hand side } \\[1 em]4x^2-11x+6-2x^2+3x &= 0&& \text{simplify left side} \\[1 em]2x^2-8x+6 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}-8x+6 = 0 $ is a quadratic equation.
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