If `x^(2) -2 sqrt(5)x +1=0`, then what is the value of `x^(5) +(1)/(x^(5))`?

Options

A. `406sqrt(5)`
B. `610sqrt(5)`
C. `408sqrt(5)`
D. `612sqrt(5)`

Answer

To solve the equation \( x^{2} - 2\sqrt{5}x + 1 = 0 \) 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:
\[ x^{2} - 2\sqrt{5}x + 1 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), where \( a = 1, b = -2\sqrt{5}, c = 1 \):
\[ x = \frac{2\sqrt{5} \pm \sqrt{(2\sqrt{5})^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (2\sqrt{5})^{2} - 4 = 20 - 4 = 16 \]
Thus, we have:
\[ x = \frac{2\sqrt{5} \pm 4}{2} \]
This gives us two possible values for \( x \):
\[ x = \sqrt{5} + 2 \quad \text{or} \quad x = \sqrt{5} - 2 \]

### Step 2: Calculate \( x + \frac{1}{x} \)
Next, we calculate \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = 2\sqrt{5} \]

### Step 3: Calculate \( x^{2} + \frac{1}{x^{2}} \)
Using the identity:
\[ x^{2} + \frac{1}{x^{2}} = (x + \frac{1}{x})^{2} - 2 \]
Substituting \( x + \frac{1}{x} = 2\sqrt{5} \):
\[ x^{2} + \frac{1}{x^{2}} = (2\sqrt{5})^{2} - 2 = 20 - 2 = 18 \]

### Step 4: Calculate \( x^{3} + \frac{1}{x^{3}} \)
Using the identity:
\[ x^{3} + \frac{1}{x^{3}} = (x + \frac{1}{x})(x^{2} + \frac{1}{x^{2}}) - (x + \frac{1}{x}) \]
Substituting the known values:
\[ x^{3} + \frac{1}{x^{3}} = (2\sqrt{5})(18) - (2\sqrt{5}) = 36\sqrt{5} - 2\sqrt{5} = 34\sqrt{5} \]

### Step 5: Calculate \( x^{5} + \frac{1}{x^{5}} \)
Using the identity:
\[ x^{5} + \frac{1}{x^{5}} = (x^{2} + \frac{1}{x^{2}})(x^{3} + \frac{1}{x^{3}}) - (x + \frac{1}{x}) \]
Substituting the known values:
\[ x^{5} + \frac{1}{x^{5}} = (18)(34\sqrt{5}) - (2\sqrt{5}) = 612\sqrt{5} - 2\sqrt{5} = 610\sqrt{5} \]

### Final Answer
Thus, the value of \( x^{5} + \frac{1}{x^{5}} \) is:
\[ \boxed{610\sqrt{5}} \]