If `x=3+sqrt8, then (x^2+1/x^2)=?`

Options

A. 34
B. 24
C. 38
D. 36

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: Calculate \( x^2 \)
Given:
\[ x = 3 + \sqrt{8} \]
First, we calculate \( x^2 \):
\[ x^2 = (3 + \sqrt{8})^2 \]
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ x^2 = 3^2 + 2 \cdot 3 \cdot \sqrt{8} + (\sqrt{8})^2 \]
Calculating each term:
\[ = 9 + 6\sqrt{8} + 8 \]
\[ = 17 + 6\sqrt{8} \]
Since \( \sqrt{8} = 2\sqrt{2} \):
\[ = 17 + 12\sqrt{2} \]

### Step 2: Calculate \( \frac{1}{x} \)
Next, we find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{3 + \sqrt{8}} \]
To rationalize the denominator, multiply the numerator and denominator by the conjugate:
\[ \frac{1}{x} = \frac{3 - \sqrt{8}}{(3 + \sqrt{8})(3 - \sqrt{8})} \]
Calculating the denominator:
\[ (3 + \sqrt{8})(3 - \sqrt{8}) = 3^2 - (\sqrt{8})^2 = 9 - 8 = 1 \]
Thus:
\[ \frac{1}{x} = 3 - \sqrt{8} \]

### Step 3: Calculate \( \frac{1}{x^2} \)
Now, we find \( \frac{1}{x^2} \):
\[ \frac{1}{x^2} = \left(3 - \sqrt{8}\right)^2 \]
Using the same formula:
\[ = 3^2 - 2 \cdot 3 \cdot \sqrt{8} + (\sqrt{8})^2 \]
Calculating:
\[ = 9 - 6\sqrt{8} + 8 \]
\[ = 17 - 6\sqrt{8} \]
Substituting \( \sqrt{8} = 2\sqrt{2} \):
\[ = 17 - 12\sqrt{2} \]

### Step 4: Combine \( x^2 \) and \( \frac{1}{x^2} \)
Now we can find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = (17 + 12\sqrt{2}) + (17 - 12\sqrt{2}) \]
The \( 12\sqrt{2} \) terms cancel out:
\[ = 17 + 17 = 34 \]

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