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. 3
• B. 1
• C. 2
• D. 4

Hint:

When solving equations with products of terms, set each factor equal to zero and solve separately. Always check for extraneous solutions when dealing with square roots.

Answer 1

The Correct Option is C

Step 1: The equation is a product of two factors. For the equation to be true, at least one of the factors must equal zero.
Step 2: Set each factor equal to zero and solve: 1. \( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 = 0 \) 2. \( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 = 0 \) 
Step 3: Solve each resultant equation for \( x \). Verify that all solutions adhere to the problem's constraints. 
Step 4: Solving both equations yields 2 distinct values for \( x \). Therefore, the correct answer is (3).