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

Options

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

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 properties of the roots of a quadratic equation.

### Step 1: Identify the coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 4 \)
- \( b = 3 \)
- \( c = 7 \)

### Step 2: Use the relationships of the roots
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 3: Calculate the sum of the roots
Using the formula for the sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{3}{4} \]

### Step 4: Calculate the product of the roots
Using the formula for the product of the roots:
\[ \alpha \beta = \frac{c}{a} = \frac{7}{4} \]

### Step 5: Find \( \frac{1}{\alpha} + \frac{1}{\beta} \)
We can express \( \frac{1}{\alpha} + \frac{1}{\beta} \) in terms of \( \alpha + \beta \) and \( \alpha \beta \):
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \]

### Step 6: Substitute the values
Now substitute the values we found:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-\frac{3}{4}}{\frac{7}{4}} = \frac{-3}{4} \times \frac{4}{7} = \frac{-3}{7} \]

### Final Answer
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is:
\[ \frac{-3}{7} \]