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Question

Solve the quadratic equation $$\dfrac{x}{x+1}+\dfrac{x+1}{x}=\dfrac{34}{15}$$ by factorization method.


Solution

$$\dfrac{x}{x+1}+\dfrac{x+1}{x}=\dfrac{34}{15}$$
$$\Rightarrow \dfrac{{x}^{2}+{\left(x+1\right)}^{2}}{x\left(x+1\right)}=\dfrac{34}{15}$$
$$\Rightarrow \dfrac{{x}^{2}+{x}^{2}+2x+1}{{x}^{2}+x}=\dfrac{34}{15}$$
$$\Rightarrow 15\left(2{x}^{2}+2x+1\right)=34{x}^{2}+34x$$
$$\Rightarrow 30{x}^{2}+30x+15-34{x}^{2}-34x=0$$
$$\Rightarrow -4{x}^{2}-4x+15=0$$
$$\Rightarrow 4{x}^{2}+4x-15=0$$
$$\Rightarrow 4{x}^{2}+10x-6x-15=0$$
$$\Rightarrow 2x\left(2x+5\right)-3\left(2x+5\right)=0$$
$$\Rightarrow \left(2x+5\right)\left(2x-3\right)=0$$
$$\therefore x=\dfrac{-5}{2},\dfrac{3}{2}$$

Mathematics

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