If `alpha` and `beta` are the roots of the equation `3x^(2)+8x+2=0` then `((1)/(alpha)+(1)/(beta))=?`

Options

A. `(-3)/(8)`
B. `(2)/(3)`
C. -4
D. 4

Explanation

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 \(3x^2 + 8x + 2 = 0\).

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is \(3x^2 + 8x + 2 = 0\). Here, we can identify:
- \(A = 3\)
- \(B = 8\)
- \(C = 2\)

2. **Use Vieta's Formulas**:
According to Vieta's formulas:
- The sum of the roots \(\alpha + \beta = -\frac{B}{A}\)
- The product of the roots \(\alpha \beta = \frac{C}{A}\)

Substituting the values:
- \(\alpha + \beta = -\frac{8}{3}\)
- \(\alpha \beta = \frac{2}{3}\)

3. **Find \(\frac{1}{\alpha} + \frac{1}{\beta}\)**:
We can express \(\frac{1}{\alpha} + \frac{1}{\beta}\) as follows:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]
Substituting the values from Vieta's formulas:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{8}{3}}{\frac{2}{3}} \]

4. **Simplify the expression**:
To simplify \(\frac{-\frac{8}{3}}{\frac{2}{3}}\):
\[ = -\frac{8}{3} \times \frac{3}{2} = -\frac{8 \times 3}{3 \times 2} = -\frac{8}{2} = -4 \]

5. **Final Answer**:
Therefore, the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is \(-4\).

### Final Result:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = -4 \]