The number of solutions of the equation ((9)/(x)-9x+2)((2)/(x)-7x+3)=0 is :

Question

The number of solutions of the equation

$(\frac{9}{x}-\frac{9}{\sqrt{x}}+2)(\frac{2}{x}-\frac{7}{\sqrt{x}}+3)=0$ is :

Accepted Answer

Let $\alpha = \dfrac{1}{\sqrt x}$ ($x>0$). Then $\dfrac1x = \alpha^2$, and the equation becomes
$$(9\alpha^2-9\alpha+2)(2\alpha^2-7\alpha+3)=0 \implies (3\alpha-2)(3\alpha-1)(\alpha-3)(2\alpha-1)=0$$

$$\alpha = \frac13, \frac12, \frac23, 3$$

Correspondingly, $x = \dfrac{1}{\alpha^2} = 9,\ 4,\ \dfrac94,\ \dfrac19$ --- all four values are valid ($x>0$).

Number of solutions = 4.

Solutions · 1