If `alpha and beta` are the roots of the equation `x^(2) -px +q =0`, then the value of αβ is

Options

A. `log(1-px+qx^(2))`
B. `log(1+px-qx^(2))`
C. `log(1+px+qx^(2))`
D. None of these

Explanation

To find the value of \( \alpha \beta \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - px + q = 0 \), we can use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.

### Step-by-Step Solution:

1. **Identify the Coefficients**:
The given quadratic equation is \( x^2 - px + q = 0 \). Here, we can identify the coefficients:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -p \) (coefficient of \( x \))
- \( c = q \) (constant term)

2. **Apply Vieta's Formulas**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

3. **Calculate the Product of the Roots**:
Using the formula for the product of the roots:
\[ \alpha \beta = \frac{c}{a} = \frac{q}{1} = q \]

4. **Conclusion**:
Therefore, the value of \( \alpha \beta \) is \( q \).

### Final Answer:
\[ \alpha \beta = q \]