Solve each of the following quadratic equations:

Question:

Solve each of the following quadratic equations:

$\frac{x}{x+1}+\frac{x+1}{x}=2 \frac{4}{15}, \quad x \neq 0,-1$

 

Solution:

$\frac{x}{x+1}+\frac{x+1}{x}=2 \frac{4}{15}, \quad x \neq 0, \quad-1$

$\frac{x}{x+1}+\frac{x+1}{x}=2 \frac{4}{15}, \quad x \neq 0,-1$

$\Rightarrow \frac{x^{2}+(x+1)^{2}}{x(x+1)}=\frac{34}{15}$

$\Rightarrow \frac{2 x^{2}+2 x+1}{x^{2}+x}=\frac{34}{15}$

$\Rightarrow 30 x^{2}+30 x+15=34 x^{2}+34 x$

$\Rightarrow 4 x^{2}+4 x-15=0$

$\Rightarrow 4 x^{2}+10 x-6 x-15=0$

$\Rightarrow 2 x(2 x+5)-3(2 x+5)=0$

$\Rightarrow(2 x+5)(2 x-3)=0$

$\Rightarrow 2 x+5=0$ or $2 x-3=0$

$\Rightarrow x=-\frac{5}{2}$ or $x=\frac{3}{2}$

Hence, $-\frac{5}{2}$ and $\frac{3}{2}$ are the roots of the given equation.

 

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