If `alpha and beta` roots of the equation `x^(2) + bx + c = 0` then what is the value of `alpha^(-1) + beta^(-1)` ?

Options

A. `-(b)/(c )`
B. `(b)/(c )`
C. `(c )/(b)`
D. `-(c )/(b)`

Answer

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

### Step 1: Use the relationship of roots
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -\frac{b}{1} = -b \)
- The product of the roots \( \alpha \beta = \frac{c}{1} = c \)

### Step 2: Express \( \alpha^{-1} + \beta^{-1} \)
We can rewrite \( \alpha^{-1} + \beta^{-1} \) using the formula:
\[ \alpha^{-1} + \beta^{-1} = \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

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

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

### Final Answer
The answer is \( -\frac{b}{c} \).

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