If `x ^(2) + (1)/(x ^(2)) = 38,` then what is the value of `( x - (1)/(x) ) ` ?

Options

A. 9
B. 6
C. 5
D. 4

Answer

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

1. **Given Equation**:
\[ x^2 + \frac{1}{x^2} = 38 \]

2. **Rearranging the Equation**:
We can subtract 2 from both sides:
\[ x^2 + \frac{1}{x^2} - 2 = 38 - 2 \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} - 2 = 36 \]

3. **Recognizing the Identity**:
The left side can be rewritten using the identity:
\[ a^2 + b^2 - 2ab = (a - b)^2 \]
where \( a = x \) and \( b = \frac{1}{x} \). Thus, we have:
\[ \left( x - \frac{1}{x} \right)^2 = 36 \]

4. **Taking the Square Root**:
Now, we take the square root of both sides:
\[ x - \frac{1}{x} = \pm 6 \]

5. **Final Answer**:
Since the question asks for the value of \( x - \frac{1}{x} \), we can conclude that:
\[ x - \frac{1}{x} = 6 \quad \text{or} \quad x - \frac{1}{x} = -6 \]
However, since we are looking for a positive value, we take:
\[ x - \frac{1}{x} = 6 \]