Question
$$2 \cdot \frac{x}{x+1} = 2-\frac{5}{2}x$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 105 }}{ 10 } & x_2 = -\dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 105 }}{ 10 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} 2 \cdot \frac{x}{x+1} &= 2-\frac{5}{2}x&& \text{multiply ALL terms by } \color{blue}{ (x+1)\cdot2 }. \\[1 em](x+1)\cdot2\cdot2 \cdot \frac{x}{x+1} &= (x+1)\cdot2\cdot2-(x+1)\cdot2 \cdot \frac{5}{2}x&& \text{cancel out the denominators} \\[1 em]4x &= 4x+4-(5x^2+5x)&& \text{simplify right side} \\[1 em]4x &= 4x+4-5x^2-5x&& \\[1 em]4x &= -5x^2-x+4&& \text{move all terms to the left hand side } \\[1 em]4x+5x^2+x-4 &= 0&& \text{simplify left side} \\[1 em]5x^2+5x-4 &= 0&& \\[1 em] \end{aligned} $$
$ 5x^{2}+5x-4 = 0 $ is a quadratic equation.
You can use step-by-step quadratic equation solver to see a detailed explanation on how to solve this equation.
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