If `alpha,beta` are the roots of the equation `ax^(2)+bx+c=0` then `log(a-bx+cx^(2))` is equal to

Options

A. `log a+(alpha+beta)x+(alpha^(2)+beta^(1))/(2)x^(2)+(alpha^(3)+beta^(3))/(3)x^(3)`+...
B. `log a +(alpha+beta) x+(alpha^(2)+beta^(2))/(2)x^(2)+(alpha^(3)+beta^(3))/(3)x^(3)`+…
C. `log a-(alpha+beta)x-(alpha^(2)+beta^(1))/(2)x^(2)-(alpha^(3)+beta^(3))/(3)x^(3)`+...
D. none of these

Answer

To solve the problem, we need to evaluate the expression \( \log(a - bx + cx^2) \) given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step-by-Step Solution:

1. **Identify the Roots**:
Since \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), we can use Vieta's formulas:
- \( \alpha + \beta = -\frac{b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)

2. **Rearranging the Expression**:
We start with the expression \( a - bx + cx^2 \). We can rewrite it as:
\[ a - bx + cx^2 = c\left(\frac{a}{c} - \frac{b}{c}x + x^2\right) \]
This can also be expressed as:
\[ c\left(x^2 - \frac{b}{c}x + \frac{a}{c}\right) \]

3. **Factoring the Quadratic**:
The quadratic \( x^2 - \frac{b}{c}x + \frac{a}{c} \) can be factored using its roots \( \alpha \) and \( \beta \):
\[ x^2 - (\alpha + \beta)x + \alpha \beta = (x - \alpha)(x - \beta) \]
Substituting the values from Vieta's, we have:
\[ x^2 - \left(-\frac{b}{a}\right)x + \frac{c}{a} = (x - \alpha)(x - \beta) \]

4. **Substituting Back**:
Thus, we can express \( a - bx + cx^2 \) as:
\[ a - bx + cx^2 = c\left(x - \alpha\right)\left(x - \beta\right) \]

5. **Taking the Logarithm**:
Now, we can take the logarithm of the expression:
\[ \log(a - bx + cx^2) = \log\left(c(x - \alpha)(x - \beta)\right) \]
Using the properties of logarithms, we can split this into:
\[ \log(c) + \log(x - \alpha) + \log(x - \beta) \]

6. **Final Expression**:
Therefore, the final expression for \( \log(a - bx + cx^2) \) is:
\[ \log(c) + \log(x - \alpha) + \log(x - \beta) \]

### Final Answer:
\[ \log(a - bx + cx^2) = \log(c) + \log(x - \alpha) + \log(x - \beta) \]