If `alpha,beta` are roots of the equation `x^2+3x+7=0,` then `1//alpha+1beta` is equal to `7//3` (b) `-7//3` (c) `3//7` (d) `-3//7`

Options

A. `7//3`
B. `-7//3`
C. `3//7`
D. `-3//7`

Answer

To solve the problem, we need to find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \), where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 + 3x + 7 = 0 \).

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation**:
The given quadratic equation is \( x^2 + 3x + 7 = 0 \).
Here, \( a = 1 \), \( b = 3 \), and \( c = 7 \).

2. **Use Vieta's formulas**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{3}{1} = -3 \).
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{7}{1} = 7 \).

3. **Calculate \( \frac{1}{\alpha} + \frac{1}{\beta} \)**:
We can express \( \frac{1}{\alpha} + \frac{1}{\beta} \) using the sum and product of the roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]
Substituting the values from Vieta's formulas:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-3}{7} \]

4. **Conclusion**:
Therefore, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is \( -\frac{3}{7} \).

### Final Answer:
The correct option is (d) \( -\frac{3}{7} \).