If `(x^2+1/x^2)=6` then the value of `(x-1/x)` is

Options

A. 2
B. 3
C. -2
D. both(a) and ©

Answer

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

### Step 1: Use the identity
We know that:
\[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \]
This means we can express \( x^2 + \frac{1}{x^2} \) in terms of \( x - \frac{1}{x} \).

### Step 2: Substitute the known value
Given \( x^2 + \frac{1}{x^2} = 6 \), we can substitute this into the identity:
\[ 6 = \left(x - \frac{1}{x}\right)^2 + 2 \]

### Step 3: Rearrange the equation
Subtract 2 from both sides:
\[ 6 - 2 = \left(x - \frac{1}{x}\right)^2 \]
\[ 4 = \left(x - \frac{1}{x}\right)^2 \]

### Step 4: Take the square root
Now, we take the square root of both sides:
\[ x - \frac{1}{x} = \pm 2 \]

### Step 5: Conclusion
Thus, the values of \( x - \frac{1}{x} \) are:
\[ x - \frac{1}{x} = 2 \quad \text{or} \quad x - \frac{1}{x} = -2 \]

### Final Answer
The values of \( x - \frac{1}{x} \) are \( 2 \) and \( -2 \).

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