If `(x - (1)/(x)) = 6`, then `(x^(2) + (1)/(x^(2))) = ?`

Options

A. 36
B. 38
C. 32
D. `36 (1)/(36)`

Answer

To solve the equation \( x - \frac{1}{x} = 6 \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ x - \frac{1}{x} = 6 \]
Now, we square both sides:
\[ \left( x - \frac{1}{x} \right)^2 = 6^2 \]

### Step 2: Expand the left side using the identity
Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \), we can expand the left side:
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 36 \]
This simplifies to:
\[ x^2 - 2 + \frac{1}{x^2} = 36 \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 36 + 2 \]
\[ x^2 + \frac{1}{x^2} = 38 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{38} \]

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