If `x ^(2) - 5x + 1 = 0,` then `x ^(5) + (1)/( x ^(5))` is equal to

Options

A. 2424
B. 3232
C. 2525
D. None

Answer

To solve the equation \( x^2 - 5x + 1 = 0 \) and find the value of \( x^5 + \frac{1}{x^5} \), we can follow these steps:

### Step 1: Solve the Quadratic Equation
We start with the quadratic equation:
\[ x^2 - 5x + 1 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -5, c = 1 \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 4}}{2} = \frac{5 \pm \sqrt{21}}{2} \]

### Step 2: Find \( x + \frac{1}{x} \)
Next, we find \( x + \frac{1}{x} \). We know:
\[ x + \frac{1}{x} = \frac{x^2 + 1}{x} \]
From the quadratic equation, we can express \( x^2 \) as:
\[ x^2 = 5x - 1 \]
Thus:
\[ x + \frac{1}{x} = \frac{(5x - 1) + 1}{x} = \frac{5x}{x} = 5 \]

### Step 3: Find \( x^2 + \frac{1}{x^2} \)
Using the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Substituting \( x + \frac{1}{x} = 5 \):
\[ x^2 + \frac{1}{x^2} = 5^2 - 2 = 25 - 2 = 23 \]

### Step 4: Find \( x^3 + \frac{1}{x^3} \)
Using the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the known values:
\[ x^3 + \frac{1}{x^3} = 5 \cdot 23 - 5 = 115 - 5 = 110 \]

### Step 5: Find \( x^5 + \frac{1}{x^5} \)
Using the identity:
\[ x^5 + \frac{1}{x^5} = \left( x^3 + \frac{1}{x^3} \right) \left( x^2 + \frac{1}{x^2} \right) - \left( x + \frac{1}{x} \right) \]
Substituting the known values:
\[ x^5 + \frac{1}{x^5} = 110 \cdot 23 - 5 \]
Calculating:
\[ 110 \cdot 23 = 2530 \]
Thus:
\[ x^5 + \frac{1}{x^5} = 2530 - 5 = 2525 \]

### Final Answer
The value of \( x^5 + \frac{1}{x^5} \) is:
\[ \boxed{2525} \]