If ` x^(4) + x^(-4) = 194 , x gt 0 ` then the value of ` ( x - 2) ^(2)` is

Options

A. 1
B. 6
C. 2
D. 3

Answer

To solve the equation \( x^4 + x^{-4} = 194 \) and find the value of \( (x - 2)^2 \), we can follow these steps:

### Step 1: Rewrite the equation
Given:
\[ x^4 + x^{-4} = 194 \]
We can rewrite \( x^{-4} \) as \( \frac{1}{x^4} \):
\[ x^4 + \frac{1}{x^4} = 194 \]

### Step 2: Use a substitution
Let \( y = x^2 + \frac{1}{x^2} \). We know that:
\[ x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 \]
Thus, we can rewrite the equation as:
\[ y^2 - 2 = 194 \]

### Step 3: Solve for \( y \)
Adding 2 to both sides gives:
\[ y^2 = 196 \]
Taking the square root of both sides:
\[ y = \sqrt{196} = 14 \]

### Step 4: Relate \( y \) back to \( x \)
Now we have:
\[ x^2 + \frac{1}{x^2} = 14 \]

### Step 5: Use another substitution
Let \( z = x + \frac{1}{x} \). We know that:
\[ x^2 + \frac{1}{x^2} = z^2 - 2 \]
Thus:
\[ z^2 - 2 = 14 \]
Adding 2 to both sides gives:
\[ z^2 = 16 \]
Taking the square root of both sides:
\[ z = 4 \quad (\text{since } x > 0) \]

### Step 6: Find \( x \)
Now we can relate \( z \) back to \( x \):
\[ x + \frac{1}{x} = 4 \]
Multiplying both sides by \( x \) gives:
\[ x^2 - 4x + 1 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]
Since \( x > 0 \), we take:
\[ x = 2 + \sqrt{3} \]

### Step 8: Calculate \( (x - 2)^2 \)
Now we need to find:
\[ (x - 2)^2 = (2 + \sqrt{3} - 2)^2 = (\sqrt{3})^2 = 3 \]

### Final Answer
Thus, the value of \( (x - 2)^2 \) is:
\[ \boxed{3} \]