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

Options

A. `(-b)/(a)`
B. `(-b)/(c ^(2))`
C. `(-b)/(c)`
D. `(- 2b)/(0)`

Answer

To find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Use the formula for the sum and product of 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 2: Write the expression for \( \frac{1}{\alpha} + \frac{1}{\beta} \)
We can rewrite the expression \( \frac{1}{\alpha} + \frac{1}{\beta} \) using the formula for the sum of fractions:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

### Step 3: Substitute the values from Vieta's formulas
Now, substituting the values we found in Step 1 into the expression:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{b}{a}}{\frac{c}{a}} \]

### Step 4: Simplify the expression
When we simplify the expression, we find:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-b/a}{c/a} = \frac{-b}{c} \]

### Conclusion
Thus, the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = -\frac{b}{c} \]