If `sqrt(x) - (1)/(sqrt(x)) = 3sqrt(2)`, then `x^(2) + (1)/(x^(2))` is equal to :

Options

A. A)402
B. B)324
C. C)326
D. D)398

Answer

To solve the equation \( \sqrt{x} - \frac{1}{\sqrt{x}} = 3\sqrt{2} \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Let \( y = \sqrt{x} \)
This means we can rewrite the equation as:
\[ y - \frac{1}{y} = 3\sqrt{2} \]

### Step 2: Multiply both sides by \( y \)
To eliminate the fraction, we multiply both sides by \( y \):
\[ y^2 - 1 = 3\sqrt{2}y \]

### Step 3: Rearrange the equation
Rearranging gives us a quadratic equation:
\[ y^2 - 3\sqrt{2}y - 1 = 0 \]

### Step 4: Use the quadratic formula
To solve for \( y \), we can use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -3\sqrt{2} \), and \( c = -1 \):
\[ y = \frac{3\sqrt{2} \pm \sqrt{(3\sqrt{2})^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (3\sqrt{2})^2 = 18 \quad \text{and} \quad -4 \cdot 1 \cdot (-1) = 4 \]
Thus,
\[ b^2 - 4ac = 18 + 4 = 22 \]
Now substituting back:
\[ y = \frac{3\sqrt{2} \pm \sqrt{22}}{2} \]

### Step 5: Find \( x \)
Since \( y = \sqrt{x} \), we have:
\[ \sqrt{x} = \frac{3\sqrt{2} \pm \sqrt{22}}{2} \]
Squaring both sides gives:
\[ x = \left(\frac{3\sqrt{2} \pm \sqrt{22}}{2}\right)^2 \]

### Step 6: Calculate \( x^2 + \frac{1}{x^2} \)
We can use the identity:
\[ x^2 + \frac{1}{x^2} = \left(y^2 + \frac{1}{y^2}\right) \]
We know:
\[ y^2 + \frac{1}{y^2} = (y - \frac{1}{y})^2 + 2 \]
From our earlier equation, we have:
\[ y - \frac{1}{y} = 3\sqrt{2} \]
Thus:
\[ (y - \frac{1}{y})^2 = (3\sqrt{2})^2 = 18 \]
So:
\[ y^2 + \frac{1}{y^2} = 18 + 2 = 20 \]

### Step 7: Final calculation
Thus, we have:
\[ x^2 + \frac{1}{x^2} = 20 \]

### Conclusion
The final answer is:
\[ \boxed{20} \]