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 \( x + \frac{1}{x} \)
Now that we have \( x = 1 \), we can calculate:
\[ x + \frac{1}{x} = 1 + \frac{1}{1} = 1 + 1 = 2 \]

### Step 3: Square \( x + \frac{1}{x} \)
Next, we square the result from Step 2:
\[ \left(x + \frac{1}{x}\right)^2 = 2^2 = 4 \]
Expanding the left side:
\[ x^2 + 2 + \frac{1}{x^2} = 4 \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} = 4 - 2 = 2 \]

### Step 4: Square \( x^2 + \frac{1}{x^2} \)
Now we square the result from Step 3:
\[ \left(x^2 + \frac{1}{x^2}\right)^2 = 2^2 = 4 \]
Expanding this gives:
\[ x^4 + 2 + \frac{1}{x^4} = 4 \]
This simplifies to:
\[ x^4 + \frac{1}{x^4} = 4 - 2 = 2 \]

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

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