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 find the expression for \( \log(a - bx + cx^2) \) 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**:
Given the quadratic equation \( ax^2 + bx + c = 0 \), the roots \( \alpha \) and \( \beta \) can be expressed using Vieta's formulas:
- Sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots: \( \alpha \beta = \frac{c}{a} \)

2. **Rewrite the Expression**:
We need to rewrite the expression \( a - bx + cx^2 \) in a form that incorporates the roots:
\[ a - bx + cx^2 = a + c x^2 - b x \]

3. **Factor out \( a \)**:
We can factor \( a \) out of the expression:
\[ a - bx + cx^2 = a \left(1 - \frac{b}{a}x + \frac{c}{a}x^2\right) \]
Substituting the values from Vieta's formulas, we have:
\[ = a \left(1 + (\alpha + \beta)x + \alpha \beta x^2\right) \]

4. **Apply the Logarithm**:
Now we can apply the logarithm:
\[ \log(a - bx + cx^2) = \log\left(a \left(1 + (\alpha + \beta)x + \alpha \beta x^2\right)\right) \]

5. **Use Logarithm Properties**:
Using the property of logarithms \( \log(mn) = \log(m) + \log(n) \):
\[ \log(a - bx + cx^2) = \log(a) + \log\left(1 + (\alpha + \beta)x + \alpha \beta x^2\right) \]

6. **Expand the Logarithm**:
The logarithm \( \log(1 + u) \) can be expanded using Taylor series for small \( u \):
\[ \log(1 + u) \approx u - \frac{u^2}{2} + \frac{u^3}{3} - \ldots \]
Here, \( u = (\alpha + \beta)x + \alpha \beta x^2 \).

7. **Final Expression**:
Thus, we can write:
\[ \log(a - bx + cx^2) = \log(a) + \left((\alpha + \beta)x + \alpha \beta x^2\right) - \frac{1}{2}\left((\alpha + \beta)x + \alpha \beta x^2\right)^2 + \ldots \]

### Conclusion:
The final expression for \( \log(a - bx + cx^2) \) is:
\[ \log(a) + \log\left(1 + (\alpha + \beta)x + \alpha \beta x^2\right) \]