In sub - parts (i) to (x) choose the correct option and in sub - parts (xi) to (xv) , answer the questions an instructed.
Given that `alphaandbeta` are the roots equation `x^(2)+x+4` , then the value of `(alpha)/(beta)+(beta)/(alpha)` is

Options

A. `(57)/(4)`
B. `-(57)/(4)`
C. `(49)/(4)`
D. `(75)/(4)`

Answer

To solve the problem, we need to find the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) given that \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 + x + 4 = 0\).

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is \(x^2 + x + 4 = 0\). Here, we can identify:
- \(a = 1\)
- \(b = 1\)
- \(c = 4\)

2. **Use Vieta's formulas**:
According to Vieta's formulas, for a quadratic equation \(ax^2 + bx + c = 0\):
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

Therefore, we calculate:
\[ \alpha + \beta = -\frac{1}{1} = -1 \]
\[ \alpha \beta = \frac{4}{1} = 4 \]

3. **Express \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)**:
We can rewrite \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) as:
\[ \frac{\alpha^2 + \beta^2}{\alpha \beta} \]

4. **Calculate \(\alpha^2 + \beta^2\)**:
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (-1)^2 - 2 \cdot 4 = 1 - 8 = -7 \]

5. **Substitute values into the expression**:
Now we substitute \(\alpha^2 + \beta^2\) and \(\alpha \beta\) into our expression:
\[ \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{-7}{4} \]

6. **Final answer**:
Thus, the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) is:
\[ \boxed{-\frac{7}{4}} \]