If `x ^(2) - 2x + 1 = 0`, then the value of `x ^(4) + (1)/(x ^(4))` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( x^2 - 2x + 1 = 0 \) and find the value of \( x^4 + \frac{1}{x^4} \), we can follow these steps:

### Step 1: Solve the quadratic equation
The given equation is:
\[ x^2 - 2x + 1 = 0 \]
This can be factored as:
\[ (x - 1)^2 = 0 \]
Thus, we find:
\[ x - 1 = 0 \implies x = 1 \]

### Step 2: Calculate \( x^2 + \frac{1}{x^2} \)
Now that we have \( x = 1 \), we can find \( x^2 + \frac{1}{x^2} \):
\[ x^2 = 1^2 = 1 \quad \text{and} \quad \frac{1}{x^2} = \frac{1}{1^2} = 1 \]
So,
\[ x^2 + \frac{1}{x^2} = 1 + 1 = 2 \]

### Step 3: Calculate \( x^4 + \frac{1}{x^4} \)
Next, we need to find \( x^4 + \frac{1}{x^4} \). We can use the identity:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Substituting \( x^2 + \frac{1}{x^2} = 2 \):
\[ x^4 + \frac{1}{x^4} = 2^2 - 2 = 4 - 2 = 2 \]

### Final Answer
Thus, the value of \( x^4 + \frac{1}{x^4} \) is:
\[ \boxed{2} \]