If `x^(2)-4x+1=0`, then what is the value of `(x^(6)+x^(-6))` ?

Options

A. 2786
B. 2702
C. 2716
D. 2744

Answer

To solve the equation \( x^2 - 4x + 1 = 0 \) and find the value of \( x^6 + x^{-6} \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the quadratic equation:
\[ x^2 - 4x + 1 = 0 \]
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -4 \), and \( c = 1 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12 \]
Now substituting back into the quadratic formula:
\[ x = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]

### Step 2: Find \( x + x^{-1} \)
Next, we can find \( x + x^{-1} \). We know:
\[ x + \frac{1}{x} = \frac{x^2 + 1}{x} \]
From the original equation, we can divide by \( x \) (assuming \( x \neq 0 \)):
\[ x - 4 + \frac{1}{x} = 0 \implies x + \frac{1}{x} = 4 \]

### Step 3: Find \( x^2 + x^{-2} \)
Using the identity:
\[ (x + x^{-1})^2 = x^2 + 2 + x^{-2} \]
We can rearrange it to find \( x^2 + x^{-2} \):
\[ x^2 + x^{-2} = (x + x^{-1})^2 - 2 = 4^2 - 2 = 16 - 2 = 14 \]

### Step 4: Find \( x^4 + x^{-4} \)
Now, we can find \( x^4 + x^{-4} \) using:
\[ (x^2 + x^{-2})^2 = x^4 + 2 + x^{-4} \]
Rearranging gives:
\[ x^4 + x^{-4} = (x^2 + x^{-2})^2 - 2 = 14^2 - 2 = 196 - 2 = 194 \]

### Step 5: Find \( x^6 + x^{-6} \)
Finally, we can find \( x^6 + x^{-6} \) using:
\[ x^6 + x^{-6} = (x^4 + x^{-4})(x^2 + x^{-2}) - (x^2 + x^{-2}) \]
Substituting the values we found:
\[ x^6 + x^{-6} = 194 \cdot 14 - 14 \]
Calculating:
\[ 194 \cdot 14 = 2716 \]
So:
\[ x^6 + x^{-6} = 2716 - 14 = 2702 \]

### Final Answer
Thus, the value of \( x^6 + x^{-6} \) is:
\[ \boxed{2702} \]