The quadratic equation whose one root is `(3+sqrt(5))/(2-sqrt(5))` is

Options

A. `x^(2)+22x+4=0`
B. `x^(2)+22x-4=0`
C. `x^(2)+11x+8=0`
D. `x^(2)+11x-8=0`

Answer

To find the quadratic equation whose one root is given as \( \alpha = \frac{3 + \sqrt{5}}{2 - \sqrt{5}} \), we will follow these steps:

### Step 1: Simplify the root
We start by simplifying the expression for \( \alpha \):
\[ \alpha = \frac{3 + \sqrt{5}}{2 - \sqrt{5}} \]
To simplify, we will rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, which is \( 2 + \sqrt{5} \):
\[ \alpha = \frac{(3 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})} \]

### Step 2: Calculate the denominator
The denominator simplifies as follows:
\[ (2 - \sqrt{5})(2 + \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \]

### Step 3: Calculate the numerator
Now, we calculate the numerator:
\[ (3 + \sqrt{5})(2 + \sqrt{5}) = 3 \cdot 2 + 3 \cdot \sqrt{5} + \sqrt{5} \cdot 2 + \sqrt{5} \cdot \sqrt{5} = 6 + 3\sqrt{5} + 2\sqrt{5} + 5 = 11 + 5\sqrt{5} \]

### Step 4: Combine the results
Now, substituting back into the expression for \( \alpha \):
\[ \alpha = \frac{11 + 5\sqrt{5}}{-1} = -11 - 5\sqrt{5} \]

### Step 5: Find the second root
The second root \( \beta \) will be the conjugate of \( \alpha \):
\[ \beta = -11 + 5\sqrt{5} \]

### Step 6: Form the quadratic equation
Using the roots \( \alpha \) and \( \beta \), we can form the quadratic equation. The sum of the roots \( \alpha + \beta \) is:
\[ \alpha + \beta = (-11 - 5\sqrt{5}) + (-11 + 5\sqrt{5}) = -22 \]
The product of the roots \( \alpha \beta \) is:
\[ \alpha \beta = (-11 - 5\sqrt{5})(-11 + 5\sqrt{5}) = (-11)^2 - (5\sqrt{5})^2 = 121 - 125 = -4 \]

### Step 7: Write the quadratic equation
The general form of a quadratic equation is given by:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - (-22)x + (-4) = 0 \]
This simplifies to:
\[ x^2 + 22x - 4 = 0 \]

### Final Answer
The required quadratic equation is:
\[ x^2 + 22x - 4 = 0 \]