If ` alpha and beta` are the roots of ` 4x^(2) + 3x +7 =0` then the value of ` 1/alpha + 1/beta` is

Options

A. `4/7`
B. `-3/7`
C. `3/7`
D. `- 3/4`

Answer

To find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 4x^2 + 3x + 7 = 0 \), we can use the relationships between the roots and the coefficients of the polynomial.

### Step-by-step Solution:

1. **Identify the coefficients**: The given quadratic equation is \( 4x^2 + 3x + 7 = 0 \). Here, \( a = 4 \), \( b = 3 \), and \( c = 7 \).

2. **Use Vieta's formulas**: According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{3}{4} \).
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{7}{4} \).

3. **Express \( \frac{1}{\alpha} + \frac{1}{\beta} \)**: We know that:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

4. **Substitute the values**: Now we substitute the values we found using Vieta's formulas:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{3}{4}}{\frac{7}{4}} \]

5. **Simplify the expression**: The \( \frac{4}{4} \) cancels out:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-3}{7} \]

6. **Final answer**: Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is \( -\frac{3}{7} \).