Solve each of the following quadratic equations:
Question:

Solve each of the following quadratic equations:

$\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, \quad x \neq-1, \frac{1}{3}$

 

Solution:

$\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3 x-1}, \quad x \neq-1, \frac{1}{3}$

$\Rightarrow \frac{3}{x+1}-\frac{2}{3 x-1}=\frac{1}{2}$

$\Rightarrow \frac{9 x-3-2 x-2}{(x+1)(3 x-1)}=\frac{1}{2}$

$\Rightarrow \frac{7 x-5}{3 x^{2}+2 x-1}=\frac{1}{2}$

$\Rightarrow 3 x^{2}+2 x-1=14 x-10 \quad$ (Cross multiplication)

$\Rightarrow 3 x^{2}-12 x+9=0$

$\Rightarrow x^{2}-4 x+3=0$

$\Rightarrow x^{2}-3 x-x+3=0$

$\Rightarrow x(x-3)-1(x-3)=0$

$\Rightarrow(x-3)(x-1)=0$

$\Rightarrow x-3=0$ or $x-1=0$

$\Rightarrow x=3$ or $x=1$

Hence, 1 and 3 are the roots of the given equation.

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