If `alpha, beta` are roots of the equation `x^(2) + x + 1 = 0`, then the equation whose roots are `(alpha)/(beta) and (beta)/(alpha)`, is

Explanation

To solve the problem, we need to find the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\), given that \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 + x + 1 = 0\).

### Step 1: Identify the coefficients of the original equation
The given equation is:
\[ x^2 + x + 1 = 0 \]
Here, we can identify:
- \(a = 1\)
- \(b = 1\)
- \(c = 1\)

### Step 2: Calculate the sum and product of the roots
For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \(\alpha + \beta\) and the product of the roots \(\alpha \beta\) can be calculated as follows:
- Sum of roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{1}{1} = -1 \]
- Product of roots:
\[ \alpha \beta = \frac{c}{a} = \frac{1}{1} = 1 \]

### Step 3: Form the new equation with roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\)
The new roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). We need to find their sum and product:
- Sum of the new roots:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]
- Product of the new roots:
\[ \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1 \]

### Step 4: Calculate \(\alpha^2 + \beta^2\)
We can find \(\alpha^2 + \beta^2\) using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the known values:
\[ \alpha^2 + \beta^2 = (-1)^2 - 2 \cdot 1 = 1 - 2 = -1 \]

### Step 5: Substitute the values into the sum of the new roots
Now we substitute \(\alpha^2 + \beta^2\) and \(\alpha \beta\) into the sum of the new roots:
\[ \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{-1}{1} = -1 \]

### Step 6: Form the new quadratic equation
Using the sum and product of the new roots, we can form the quadratic equation:
\[ x^2 - \text{(sum of roots)} \cdot x + \text{(product of roots)} = 0 \]
Substituting the values:
\[ x^2 - (-1)x + 1 = 0 \implies x^2 + x + 1 = 0 \]

### Conclusion
Thus, the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) is:
\[ \boxed{x^2 + x + 1 = 0} \]