Question

The number of rational roots of $81{\left(\frac{2x-5}{3x+1}\right)}^4-45{\left(\frac{2x-5}{3x+1}\right)}^2+4=0,x\ne 1/3$ is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

Put ${\left(\frac{2x-5}{3x+1}\right)}^2=t$, so that equation becomes
$81t^2-45t+4=0$
$\Rightarrow 81t^2-36t-9t+4=0$
$\Rightarrow (9t-1)(9t-4)=0$
$\Rightarrow t=\frac{1}{9},\frac{4}{9}\Rightarrow \frac{2x-5}{3x+1}=\pm \frac{1}{3},\pm \frac{2}{3}$
Thus, the given equation has four rational roots.