If `x = 3 + sqrt(8) ` then `(x^(2) + (1)/(x^(2)))=?`

Options

A. `34`
B. `56`
C. `28`
D. `63`

Answer

To solve the problem where \( x = 3 + \sqrt{8} \) and we need to find \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Find \( x + \frac{1}{x} \)

First, we need to calculate \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{3 + \sqrt{8}} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1}{x} = \frac{1 \cdot (3 - \sqrt{8})}{(3 + \sqrt{8})(3 - \sqrt{8})} \]
Calculating the denominator:
\[ (3 + \sqrt{8})(3 - \sqrt{8}) = 3^2 - (\sqrt{8})^2 = 9 - 8 = 1 \]
Thus,
\[ \frac{1}{x} = 3 - \sqrt{8} \]

Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (3 + \sqrt{8}) + (3 - \sqrt{8}) = 3 + 3 = 6 \]

### Step 2: Use the identity to find \( x^2 + \frac{1}{x^2} \)

We can use the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Substituting the value we found:
\[ x^2 + \frac{1}{x^2} = 6^2 - 2 = 36 - 2 = 34 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{34} \]