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. `3/7`

Answer

To find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) where \( \alpha \) and \( \beta \) are the roots of the equation \( 4x^2 + 3x + 7 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is:
\[ 4x^2 + 3x + 7 = 0 \]
Here, the coefficients are:
- \( A = 4 \) (coefficient of \( x^2 \))
- \( B = 3 \) (coefficient of \( x \))
- \( C = 7 \) (constant term)

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{B}{A} = -\frac{3}{4} \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{C}{A} = \frac{7}{4} \]

### Step 3: Find \( \frac{1}{\alpha} + \frac{1}{\beta} \)
We can express \( \frac{1}{\alpha} + \frac{1}{\beta} \) in terms of 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:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{3}{4}}{\frac{7}{4}} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-\frac{3}{4}}{\frac{7}{4}} = -\frac{3}{4} \times \frac{4}{7} = -\frac{3}{7} \]

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