Question

For the equation $\frac{2x}{3x^2-x+2}-\frac{7x}{3x^2+5x+2}=1$, which of the following statements(s) is/are?

• A. It has $4$ real roots.
• B. It has $2$ real roots.
• C. Sum of roots is $-3$
• D. Sum of roots is $-\frac{11}{3}$

Answer 1

Answer: (B), (C)

The correct options are
B It has $2$ real roots.
C Sum of roots is $-3$

$\frac{2x}{3x^2-x+2}-\frac{7x}{3x^2+5x+2}=1\Rightarrow \frac{2}{3x+\frac{2}{x}-1}-\frac{7}{3x+\frac{2}{x}+5}=1$
Assuming $3x+\frac{2}{x}=t$
$\Rightarrow \frac{2}{t-1}-\frac{7}{t+5}=1\Rightarrow 2t+10-7t+7=(t-1)(t+5)\Rightarrow t^2+4t-5=-5t+17\Rightarrow t^2+9t-22=0\Rightarrow (t+11)(t-2)=0\Rightarrow t=-11,2$

When $t=-11$
$3x+\frac{2}{x}=-11\Rightarrow 3x^2+11x+2=0D=121-24>0$
Two real and distinct root
Sum of roots $=-\frac{11}{3}$

When $t=2$
$3x+\frac{2}{x}=2\Rightarrow 3x^2-2x+2=0D=4-24<0$
No real roots.
Sum of roots $=\frac{2}{3}$

Therefore,
Sum of the roots $=-\frac{11}{3}+\frac{2}{3}=-3$
The given equation has $2$ real roots.