Question

The number of solutions of the equation $\left(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2\right)\left(\frac{2}{x} - \frac{7}{\sqrt{x}} + 3\right) = 0$ is:

• A. 2
• B. 3
• C. 1
• D. 4

Answer 1

Answer: (D)

4

Answer 2

Let $y = \frac{1}{\sqrt{x}}$. The equation transforms to $(9y^2 - 9y + 2)(2y^2 - 7y + 3) = 0$. The condition $x>0$ implies $y>0$. Solving $9y^2 - 9y + 2 = 0$ yields $y = \frac{1}{3}, \frac{2}{3}$. Solving $2y^2 - 7y + 3 = 0$ yields $y = \frac{1}{2}, 3$. All four $y$ values are positive. Each positive $y$ corresponds to a unique positive $x = 1/y^2$. Thus, there are 4 distinct solutions.