Find the equation whose roots are `(alpha)/(beta) and (beta)/(alpha)`, where `alpha and beta` are the roots of the equation `x^(2)+2x+3=0`.

Answer

To find the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\), where \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 + 2x + 3 = 0\), we can follow these steps:

### Step 1: Find the roots \(\alpha\) and \(\beta\)
The roots of the quadratic equation \(x^2 + 2x + 3 = 0\) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 2\), and \(c = 3\).

Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \]
Since the discriminant is negative, the roots are complex:
\[ x = \frac{-2 \pm \sqrt{-8}}{2 \cdot 1} = \frac{-2 \pm 2i\sqrt{2}}{2} = -1 \pm i\sqrt{2} \]
Thus, the roots are:
\[ \alpha = -1 + i\sqrt{2}, \quad \beta = -1 - i\sqrt{2} \]

### Step 2: Calculate \(\alpha + \beta\) and \(\alpha \beta\)
Using Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} = -2 \]
\[ \alpha \beta = \frac{c}{a} = 3 \]

### Step 3: Find the sum and product of the new roots
The new roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\).

**Sum of the new roots:**
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]
Using the identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\):
\[ \alpha^2 + \beta^2 = (-2)^2 - 2 \cdot 3 = 4 - 6 = -2 \]
Thus,
\[ \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{-2}{3} \]

**Product of the new roots:**
\[ \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1 \]

### Step 4: Form the new quadratic equation
Using the sum and product of the roots, we can write the quadratic equation:
\[ x^2 - \left(\text{sum of roots}\right)x + \text{product of roots} = 0 \]
Substituting the values:
\[ x^2 - \left(-\frac{2}{3}\right)x + 1 = 0 \]
This simplifies to:
\[ x^2 + \frac{2}{3}x + 1 = 0 \]
To eliminate the fraction, multiply the entire equation by 3:
\[ 3x^2 + 2x + 3 = 0 \]

### Final Answer
The required quadratic equation is:
\[ 3x^2 + 2x + 3 = 0 \]