If `(x^4+1/x^4)=34` then the value of `(x-1/x)^2` is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

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

**Step 1: Start with the equation**
Given:
\[ x^4 + \frac{1}{x^4} = 34 \]

**Step 2: Add 2 to both sides**
We add 2 to both sides of the equation:
\[ x^4 + \frac{1}{x^4} + 2 = 34 + 2 \]
This simplifies to:
\[ x^4 + \frac{1}{x^4} + 2 = 36 \]

**Step 3: Recognize the identity**
Notice that:
\[ x^4 + \frac{1}{x^4} + 2 = \left(x^2 + \frac{1}{x^2}\right)^2 \]
So we can rewrite the equation as:
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 36 \]

**Step 4: Take the square root**
Taking the square root of both sides gives us:
\[ x^2 + \frac{1}{x^2} = \sqrt{36} \]
Since we are dealing with squares, we only consider the positive root:
\[ x^2 + \frac{1}{x^2} = 6 \]

**Step 5: Use the identity to find \( (x - \frac{1}{x})^2 \)**
We know that:
\[ (x - \frac{1}{x})^2 = (x^2 + \frac{1}{x^2}) - 2 \]
Substituting the value we found:
\[ (x - \frac{1}{x})^2 = 6 - 2 \]
This simplifies to:
\[ (x - \frac{1}{x})^2 = 4 \]

**Step 6: Conclusion**
Thus, the value of \( (x - \frac{1}{x})^2 \) is:
\[ \boxed{4} \]

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