If ` x^(2) - 4 x + 1 = 0` then the value of ` (( x^(6) + 1)/( x^(3)))` is

Options

A. 48
B. 52
C. 55
D. 58

Answer

To solve the equation \( x^2 - 4x + 1 = 0 \) and find the value of \( \frac{x^6 + 1}{x^3} \), we can follow these steps:

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

Calculating the discriminant:
\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12 \]

Now applying the quadratic formula:
\[ x = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]

### Step 2: Find \( x + \frac{1}{x} \)
Next, we need to find \( x + \frac{1}{x} \). We can express \( \frac{1}{x} \) using the roots we found:
\[ x + \frac{1}{x} = x + \frac{1}{2 \pm \sqrt{3}} \]

To simplify \( \frac{1}{2 \pm \sqrt{3}} \), we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{2 \pm \sqrt{3}} \cdot \frac{2 \mp \sqrt{3}}{2 \mp \sqrt{3}} = \frac{2 \mp \sqrt{3}}{(2)^2 - (\sqrt{3})^2} = \frac{2 \mp \sqrt{3}}{4 - 3} = 2 \mp \sqrt{3} \]

Thus:
\[ x + \frac{1}{x} = (2 \pm \sqrt{3}) + (2 \mp \sqrt{3}) = 4 \]

### Step 3: Find \( x^3 + \frac{1}{x^3} \)
We can use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) \]
Substituting \( x + \frac{1}{x} = 4 \):
\[ x^3 + \frac{1}{x^3} = 4^3 - 3 \cdot 4 = 64 - 12 = 52 \]

### Step 4: Find \( \frac{x^6 + 1}{x^3} \)
Now, we can express \( x^6 + 1 \) as:
\[ x^6 + 1 = (x^3)^2 + 1 \]
Thus:
\[ \frac{x^6 + 1}{x^3} = x^3 + \frac{1}{x^3} \]
From our previous calculation, we found:
\[ x^3 + \frac{1}{x^3} = 52 \]

### Final Answer
Therefore, the value of \( \frac{x^6 + 1}{x^3} \) is:
\[ \boxed{52} \]