If `(x+(1)/(x))=6` , then `(x^(2)+(1)/(x^(2)))` is equal to

Options

A. 32
B. 38
C. 34
D. 44

Answer

To solve the problem, we need to find the value of \( x^2 + \frac{1}{x^2} \) given that \( x + \frac{1}{x} = 6 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = 6 \]

2. **Square both sides of the equation:**
\[ \left(x + \frac{1}{x}\right)^2 = 6^2 \]
This simplifies to:
\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 36 \]

3. **Simplify the left side:**
The term \( 2 \cdot x \cdot \frac{1}{x} \) simplifies to 2. Therefore, we have:
\[ x^2 + 2 + \frac{1}{x^2} = 36 \]

4. **Rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):**
\[ x^2 + \frac{1}{x^2} = 36 - 2 \]

5. **Calculate the final value:**
\[ x^2 + \frac{1}{x^2} = 34 \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is **34**.