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

Options

A. 8
B. 16
C. 10
D. 12

Answer

To solve the problem where \( x + \frac{1}{x} = 2\sqrt{3} \) and we need to find \( x^2 + \frac{1}{x^2} \), we can use a standard algebraic identity.

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = 2\sqrt{3} \]

2. **Use the identity for \( x^2 + \frac{1}{x^2} \):**
The identity states that:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]

3. **Substitute the value of \( x + \frac{1}{x} \) into the identity:**
\[ x^2 + \frac{1}{x^2} = \left( 2\sqrt{3} \right)^2 - 2 \]

4. **Calculate \( \left( 2\sqrt{3} \right)^2 \):**
\[ \left( 2\sqrt{3} \right)^2 = 4 \cdot 3 = 12 \]

5. **Now substitute this back into the equation:**
\[ x^2 + \frac{1}{x^2} = 12 - 2 \]

6. **Perform the subtraction:**
\[ x^2 + \frac{1}{x^2} = 10 \]

### Final Answer:
Thus, \( x^2 + \frac{1}{x^2} = 10 \).

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