If `x=3+2sqrt2`, then the value of `x^(2)+(1)/(x^(2))` is:

Options

A. 34
B. 30
C. 36
D. 32

Answer

To solve the problem, we need to find the value of \( x^2 + \frac{1}{x^2} \) given that \( x = 3 + 2\sqrt{2} \).

### Step-by-Step Solution:

1. **Find \( \frac{1}{x} \)**:
\[ x = 3 + 2\sqrt{2} \]
To find \( \frac{1}{x} \), we rationalize it:
\[ \frac{1}{x} = \frac{1}{3 + 2\sqrt{2}} \cdot \frac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} = \frac{3 - 2\sqrt{2}}{(3 + 2\sqrt{2})(3 - 2\sqrt{2})} \]
The denominator simplifies as follows:
\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 9 - 8 = 1 \]
Thus,
\[ \frac{1}{x} = 3 - 2\sqrt{2} \]

2. **Calculate \( x + \frac{1}{x} \)**:
\[ x + \frac{1}{x} = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 3 + 3 = 6 \]

3. **Square \( x + \frac{1}{x} \)**:
\[ (x + \frac{1}{x})^2 = 6^2 = 36 \]

4. **Use the identity to find \( x^2 + \frac{1}{x^2} \)**:
The identity states that:
\[ (x + \frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2 \]
Therefore,
\[ 36 = x^2 + \frac{1}{x^2} + 2 \]

5. **Solve for \( x^2 + \frac{1}{x^2} \)**:
\[ x^2 + \frac{1}{x^2} = 36 - 2 = 34 \]

### Final Answer:
The value of \( x^2 + \frac{1}{x^2} \) is \( 34 \).