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 (c)

Answer

To solve the equation \( x^2 + \frac{1}{x^2} = 6 \) 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 \( y \) such that:
\[ y = x + \frac{1}{x} \]

### Step 2: Relate \( y \) to \( x^2 + \frac{1}{x^2} \)

We know from algebra 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 \( y \):
\[ x^2 + \frac{1}{x^2} = y^2 - 2 \]

### Step 3: Substitute the given value

Given that \( x^2 + \frac{1}{x^2} = 6 \), we can substitute this into our equation:
\[ y^2 - 2 = 6 \]

### Step 4: Solve for \( y^2 \)

Now, we solve for \( y^2 \):
\[ y^2 = 6 + 2 \]
\[ y^2 = 8 \]

### Step 5: Take the square root

Next, we take the square root of both sides to find \( y \):
\[ y = \sqrt{8} \quad \text{or} \quad y = -\sqrt{8} \]
Since \( \sqrt{8} = 2\sqrt{2} \), we have:
\[ y = 2\sqrt{2} \quad \text{or} \quad y = -2\sqrt{2} \]

### Step 6: Conclusion

Thus, the value of \( x + \frac{1}{x} \) can be either:
\[ x + \frac{1}{x} = 2\sqrt{2} \quad \text{or} \quad x + \frac{1}{x} = -2\sqrt{2} \]

### Final Answer
The value of \( x + \frac{1}{x} \) is \( 2\sqrt{2} \) or \( -2\sqrt{2} \).

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