Question

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

• A. 3
• B. 1
• C. 4
• D. 2

Hint:

When dealing with square roots in equations, try substitution to simplify the expression and reduce it to a quadratic form.

Answer 1

The Correct Option is C

We are tasked with solving the given equation and determining the number of solutions for \( x \). Let us proceed step by step:

1. Problem Setup:
We are given the substitution:

\( \frac{1}{\sqrt{x}} = \alpha, \quad x > 0 \)

The equation to solve is:

\( (9\alpha^2 - 9\alpha + 2)(2\alpha^2 - 7\alpha + 3) = 0 \)

2. Factoring the Quartic Equation:
The quartic equation can be factored as:

\( (3\alpha - 2)(3\alpha - 1)(\alpha - 3)(2\alpha - 1) = 0 \)

3. Solving for \( \alpha \):
Set each factor equal to zero to find the values of \( \alpha \):

\( 3\alpha - 2 = 0 \implies \alpha = \frac{2}{3} \)

\( 3\alpha - 1 = 0 \implies \alpha = \frac{1}{3} \)

\( \alpha - 3 = 0 \implies \alpha = 3 \)

\( 2\alpha - 1 = 0 \implies \alpha = \frac{1}{2} \)

Thus, the solutions for \( \alpha \) are:

\( \alpha = \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, 3 \)

4. Substituting Back to Find \( x \):
Recall that \( \alpha = \frac{1}{\sqrt{x}} \). Solving for \( x \), we get:

\( x = \frac{1}{\alpha^2} \)

Substitute each value of \( \alpha \):

\( \alpha = \frac{1}{3} \implies x = \frac{1}{\left(\frac{1}{3}\right)^2} = 9 \)

\( \alpha = \frac{1}{2} \implies x = \frac{1}{\left(\frac{1}{2}\right)^2} = 4 \)

\( \alpha = \frac{2}{3} \implies x = \frac{1}{\left(\frac{2}{3}\right)^2} = \frac{9}{4} \)

\( \alpha = 3 \implies x = \frac{1}{3^2} = \frac{1}{9} \)

Thus, the corresponding values of \( x \) are:

\( x = 9, 4, \frac{9}{4}, \frac{1}{9} \)

5. Counting the Number of Solutions:
There are 4 distinct values of \( x \). Therefore, the total number of solutions is:

\( \text{Number of solutions} = 4 \)

Final Answer:
The number of solutions is \( \boxed{4} \).

Answer 2

Step 1: Given equation.
The given equation is:
\[ \left( \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 \right) = 0. \] This equation is a product of two factors, and for the product to be zero, at least one of the factors must be zero. So, we solve each factor separately.

Step 2: Solve the first factor.
The first factor is: \[ \frac{9}{x} - \frac{9}{\sqrt{x}} + 2 = 0. \] Let \( y = \sqrt{x} \), so \( x = y^2 \). Substituting into the equation: \[ \frac{9}{y^2} - \frac{9}{y} + 2 = 0. \] Multiply the whole equation by \( y^2 \) to clear the denominators: \[ 9 - 9y + 2y^2 = 0. \] This simplifies to: \[ 2y^2 - 9y + 9 = 0. \] Now, solve this quadratic equation using the quadratic formula: \[ y = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 2 \cdot 9}}{2 \cdot 2} = \frac{9 \pm \sqrt{81 - 72}}{4} = \frac{9 \pm \sqrt{9}}{4} = \frac{9 \pm 3}{4}. \] So, we have two solutions for \( y \): \[ y = \frac{9 + 3}{4} = 3 \quad \text{or} \quad y = \frac{9 - 3}{4} = \frac{3}{2}. \] Since \( y = \sqrt{x} \), we now solve for \( x \): \[ y = 3 \Rightarrow \sqrt{x} = 3 \Rightarrow x = 9, \] \[ y = \frac{3}{2} \Rightarrow \sqrt{x} = \frac{3}{2} \Rightarrow x = \left( \frac{3}{2} \right)^2 = \frac{9}{4}. \] Thus, the solutions from the first factor are \( x = 9 \) and \( x = \frac{9}{4} \).

Step 3: Solve the second factor.
The second factor is: \[ \frac{2}{x} - \frac{7}{\sqrt{x}} + 3 = 0. \] Again, let \( y = \sqrt{x} \), so \( x = y^2 \). Substituting into the equation: \[ \frac{2}{y^2} - \frac{7}{y} + 3 = 0. \] Multiply the whole equation by \( y^2 \) to clear the denominators: \[ 2 - 7y + 3y^2 = 0. \] This simplifies to: \[ 3y^2 - 7y + 2 = 0. \] Now, solve this quadratic equation using the quadratic formula: \[ y = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = \frac{7 \pm \sqrt{49 - 24}}{6} = \frac{7 \pm \sqrt{25}}{6} = \frac{7 \pm 5}{6}. \] So, we have two solutions for \( y \): \[ y = \frac{7 + 5}{6} = 2 \quad \text{or} \quad y = \frac{7 - 5}{6} = \frac{1}{3}. \] Since \( y = \sqrt{x} \), we now solve for \( x \): \[ y = 2 \Rightarrow \sqrt{x} = 2 \Rightarrow x = 4, \] \[ y = \frac{1}{3} \Rightarrow \sqrt{x} = \frac{1}{3} \Rightarrow x = \left( \frac{1}{3} \right)^2 = \frac{1}{9}. \] Thus, the solutions from the second factor are \( x = 4 \) and \( x = \frac{1}{9} \).

Step 4: Combine the solutions.
The solutions for \( x \) are: \[ x = 9, \frac{9}{4}, 4, \frac{1}{9}. \] Thus, there are 4 solutions in total.

Final Answer:
\[ \boxed{4}. \]