If `x^(2) + (1)/(x^(2)) = 11` , then ` x - (1)/(x)` is equal to :

Options

A. 2
B. 3
C. 5
D. 4

Answer

To solve the equation \( x^2 + \frac{1}{x^2} = 11 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Let \( y = x - \frac{1}{x} \)
We start by defining a new variable \( y \) for simplicity.

### Step 2: Express \( x^2 + \frac{1}{x^2} \) in terms of \( y \)
We know the identity:
\[ x^2 + \frac{1}{x^2} = \left( x - \frac{1}{x} \right)^2 + 2 \]
Substituting \( y \) into the equation gives us:
\[ x^2 + \frac{1}{x^2} = y^2 + 2 \]

### Step 3: Set up the equation
From the problem, we have:
\[ y^2 + 2 = 11 \]

### Step 4: Solve for \( y^2 \)
Subtract 2 from both sides:
\[ y^2 = 11 - 2 \]
\[ y^2 = 9 \]

### Step 5: Solve for \( y \)
Taking the square root of both sides gives:
\[ y = \pm 3 \]

### Step 6: Conclusion
Thus, \( x - \frac{1}{x} \) can be either 3 or -3. However, since the question does not specify a restriction on \( x \), we can conclude:
\[ x - \frac{1}{x} = 3 \quad \text{or} \quad x - \frac{1}{x} = -3 \]

For the purpose of this problem, we can take the positive value:
\[ \boxed{3} \]