If `alpha` and `beta` are the roots of the equation `ax^(2)+bc+c=0` then the sum of the roots of the equation `a^(2)x^(2)+(b^(2)-2ac)x+b^(2)-4ac=0` is

Options

A. `-(alpha^(2)-beta^(2))`
B. `(alpha+beta)^(2)-2alpha beta`
C. `alpha^(2)beta+beta alpha^(2)-4alpha beta`
D. `-(alpha^(2)+beta^(2))`

Explanation

To solve the problem, we need to find the sum of the roots of the equation \( a^2 x^2 + (b^2 - 2ac)x + (b^2 - 4ac) = 0 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \).

### Step-by-Step Solution:

1. **Identify the given equations**:
- The first equation is \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \).
- The second equation is \( a^2 x^2 + (b^2 - 2ac)x + (b^2 - 4ac) = 0 \).

2. **Find the sum of the roots for the first equation**:
- The sum of the roots \( \alpha + \beta \) is given by the formula:
\[ \alpha + \beta = -\frac{b}{a} \]

3. **Find the product of the roots for the first equation**:
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} \]

4. **Find the sum of the roots for the second equation**:
- The sum of the roots \( \gamma + \lambda \) for the second equation can be calculated using the formula:
\[ \gamma + \lambda = -\frac{(b^2 - 2ac)}{a^2} \]

5. **Substituting values**:
- Substitute \( b^2 - 2ac \) into the sum of the roots:
\[ \gamma + \lambda = -\frac{b^2 - 2ac}{a^2} = -\frac{b^2}{a^2} + \frac{2ac}{a^2} \]
- This simplifies to:
\[ \gamma + \lambda = -\frac{b^2}{a^2} + \frac{2c}{a} \]

6. **Express \( c \) in terms of \( \alpha \) and \( \beta \)**:
- From the product of the roots, we have \( c = a \alpha \beta \). Substitute this into the equation:
\[ \gamma + \lambda = -\frac{b^2}{a^2} + \frac{2a \alpha \beta}{a^2} \]
- This gives:
\[ \gamma + \lambda = -\frac{b^2}{a^2} + \frac{2 \alpha \beta}{a} \]

7. **Rewrite \( \frac{2 \alpha \beta}{a} \)**:
- We can express \( \alpha \beta \) as \( \frac{c}{a} \):
\[ \gamma + \lambda = -\frac{b^2}{a^2} + 2 \cdot \frac{c}{a^2} \]
- Thus:
\[ \gamma + \lambda = \frac{2c - b^2}{a^2} \]

8. **Final expression**:
- The final expression for the sum of the roots \( \gamma + \lambda \) is:
\[ \gamma + \lambda = \frac{2c - b^2}{a^2} \]

### Summary:
The sum of the roots of the equation \( a^2 x^2 + (b^2 - 2ac)x + (b^2 - 4ac) = 0 \) is given by:
\[ \gamma + \lambda = \frac{2c - b^2}{a^2} \]