If `sqrt(x)+(1)/(sqrt(x))=3,xgt0`, then `x^(2)(x^(2)-47)=?`

Options

A. 0
B. 2
C. `-2`
D. `-1`

Answer

To solve the equation given in the problem, let's go through the steps systematically.

Given:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 3 \quad (x > 0) \]

### Step 1: Square both sides
We start by squaring both sides of the equation to eliminate the square root:
\[ \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 = 3^2 \]
This expands to:
\[ x + 2 + \frac{1}{x} = 9 \]

### Step 2: Rearranging the equation
Now, we can rearrange the equation:
\[ x + \frac{1}{x} + 2 = 9 \]
Subtracting 2 from both sides gives:
\[ x + \frac{1}{x} = 7 \]

### Step 3: Square again
Next, we square both sides again:
\[ \left(x + \frac{1}{x}\right)^2 = 7^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 49 \]

### Step 4: Rearranging again
Rearranging gives:
\[ x^2 + \frac{1}{x^2} + 2 = 49 \]
Subtracting 2 from both sides results in:
\[ x^2 + \frac{1}{x^2} = 47 \]

### Step 5: Finding \(x^2(x^2 - 47)\)
Now, we need to find \(x^2(x^2 - 47)\):
Let \(y = x^2\). Then we can write:
\[ y(y - 47) = y^2 - 47y \]

### Step 6: Substitute \(y\)
Since we know \(x^2 + \frac{1}{x^2} = 47\), we can substitute \(y\) into the equation:
\[ y^2 - 47y = 0 \]
Factoring out \(y\):
\[ y(y - 47) = 0 \]
This gives us \(y = 0\) or \(y = 47\). Since \(x > 0\), we take \(y = 47\).

### Conclusion
Thus, the value of \(x^2(x^2 - 47)\) is:
\[ 47(47 - 47) = 47 \cdot 0 = 0 \]

### Final Answer
\[ \boxed{0} \]