If ` x + (1)/(x) = 6` , then find `x^(2) + (1)/(x^(2))` .

Options

A. 34
B. 36
C. 32
D. 38

Explanation

To solve the problem, we start with the equation given:

**Step 1:** We are given that \( x + \frac{1}{x} = 6 \).

**Step 2:** To find \( x^2 + \frac{1}{x^2} \), we will square both sides of the equation \( x + \frac{1}{x} = 6 \).

\[ \left( x + \frac{1}{x} \right)^2 = 6^2 \]

**Step 3:** Expanding the left side using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):

\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 36 \]

**Step 4:** Simplifying the equation:

Since \( 2 \cdot x \cdot \frac{1}{x} = 2 \), we have:

\[ x^2 + 2 + \frac{1}{x^2} = 36 \]

**Step 5:** Rearranging the equation to isolate \( x^2 + \frac{1}{x^2} \):

\[ x^2 + \frac{1}{x^2} = 36 - 2 \]

**Step 6:** Performing the subtraction:

\[ x^2 + \frac{1}{x^2} = 34 \]

Thus, the final answer is:

\[ \boxed{34} \]

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