Find the quadratic equation whose roots are `(3+- i sqrt5)/(2)`.

Answer

To find the quadratic equation whose roots are given as \((3 \pm i \sqrt{5})/2\), we can follow these steps:

### Step 1: Identify the roots
The roots are:
\[ \alpha = \frac{3 + i \sqrt{5}}{2}, \quad \beta = \frac{3 - i \sqrt{5}}{2} \]

### Step 2: Calculate the sum of the roots
The sum of the roots \((\alpha + \beta)\) can be calculated as follows:
\[ \alpha + \beta = \frac{3 + i \sqrt{5}}{2} + \frac{3 - i \sqrt{5}}{2} \]
Combining the fractions:
\[ = \frac{(3 + i \sqrt{5}) + (3 - i \sqrt{5})}{2} = \frac{6}{2} = 3 \]

### Step 3: Calculate the product of the roots
The product of the roots \((\alpha \beta)\) is calculated as:
\[ \alpha \beta = \left(\frac{3 + i \sqrt{5}}{2}\right) \left(\frac{3 - i \sqrt{5}}{2}\right) \]
Using the difference of squares formula:
\[ = \frac{(3)^2 - (i \sqrt{5})^2}{4} = \frac{9 - (-5)}{4} = \frac{9 + 5}{4} = \frac{14}{4} = \frac{7}{2} \]

### Step 4: Form the quadratic equation
The standard form of a quadratic equation with roots \(\alpha\) and \(\beta\) is given by:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - 3x + \frac{7}{2} = 0 \]

### Step 5: Eliminate the fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 2x^2 - 6x + 7 = 0 \]

### Final Result
The quadratic equation whose roots are \((3 \pm i \sqrt{5})/2\) is:
\[ \boxed{2x^2 - 6x + 7 = 0} \]