If `alpha and beta` are the roots of the quadratic equation `ax^(2)+bx+1`, then the value of `(1)/(alpha)+(1)/(beta)` is

Options

A. a
B. `-a`
C. b
D. `-b`

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 \(ax^2 + bx + 1 = 0\).

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression \(\frac{1}{\alpha} + \frac{1}{\beta}\). This can be rewritten using a common denominator:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

2. **Using Vieta's Formulas**:
According to Vieta's formulas for a quadratic equation \(ax^2 + bx + c = 0\):
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

In our case, since \(c = 1\), we have:
\[ \alpha + \beta = -\frac{b}{a} \quad \text{and} \quad \alpha \beta = \frac{1}{a} \]

3. **Substituting Values**:
Now we can substitute these values into our expression:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{b}{a}}{\frac{1}{a}} \]

4. **Simplifying the Expression**:
When we simplify the expression:
\[ \frac{-\frac{b}{a}}{\frac{1}{a}} = -b \]

5. **Final Result**:
Therefore, the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is:
\[ \boxed{-b} \]