The number of solutions of $\left(\dfrac{9}{x} - \dfrac{9}{\sqrt{x}} + 2\right)\left(\dfrac{2}{x} - \dfrac{7}{\sqrt{x}} + 3\right) = 0$ is
Step-by-Step Solution
Key Concept: Substitute $\alpha = 1/\sqrt{x}$ (requiring $x > 0$) to reduce each factor to a quadratic in $\alpha$, factorise, and back-substitute.
Let $\alpha = \dfrac{1}{\sqrt{x}}$, so the equation becomes
$$(9\alpha^2 - 9\alpha + 2)(2\alpha^2 - 7\alpha + 3) = 0$$
$$\Rightarrow (3\alpha-2)(3\alpha-1)(\alpha-3)(2\alpha-1) = 0.$$
So $\alpha = \dfrac{2}{3},\, \dfrac{1}{3},\, 3,\, \dfrac{1}{2}$, giving $x = \dfrac{9}{4},\, 9,\, \dfrac{1}{9},\, 4$.
All four values are positive, so there are **4 solutions**.
Correct Answer: 4