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  4. The number of rational roots of 81((2 x-5/3 x+1))4-45((2 x-5/3 x+1))2+4=0, x ≠ 1 / 3 is

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

Complex Numbers and Quadratic Equations

Solution:

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