If `alpha` and `beta` are the roots of `ax^(2)+bx+c=0`, then which of the following are the roots of the equation `ax^(2)-bx(x-1)+c(x-1)^(2)=0` ?

Options

A. `(alpha)/(alpha+1),(beta)/(beta+1)`
B. `(alpha+1)/(alpha)+(beta+1)/(beta)`
C. `(alpha)/(alpha-1),(beta)/(beta-1)`
D. `(alpha-1)/(alpha),(beta-1)/(beta)`

Answer

To solve the problem, we need to find the roots of the equation \( ax^2 - bx(x-1) + c(x-1)^2 = 0 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ ax^2 - bx(x-1) + c(x-1)^2 = 0 \]

2. **Expand the terms**:
- The term \( -bx(x-1) \) expands to \( -bx^2 + bx \).
- The term \( c(x-1)^2 \) expands to \( c(x^2 - 2x + 1) = cx^2 - 2cx + c \).

Thus, the equation becomes:
\[ ax^2 - bx^2 + bx + cx^2 - 2cx + c = 0 \]

3. **Combine like terms**:
- Combine the \( x^2 \) terms: \( (a - b + c)x^2 \).
- Combine the \( x \) terms: \( (b - 2c)x \).
- The constant term remains \( c \).

Therefore, we rewrite the equation as:
\[ (a - b + c)x^2 + (b - 2c)x + c = 0 \]

4. **Identify coefficients**:
The coefficients of the new quadratic equation are:
- Coefficient of \( x^2 \): \( A = a - b + c \)
- Coefficient of \( x \): \( B = b - 2c \)
- Constant term: \( C = c \)

5. **Use Vieta's formulas**:
From Vieta's formulas, we know:
- The sum of the roots \( P + Q = -\frac{B}{A} = -\frac{b - 2c}{a - b + c} \).
- The product of the roots \( PQ = \frac{C}{A} = \frac{c}{a - b + c} \).

6. **Substituting values**:
We know that:
- \( \alpha + \beta = -\frac{b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)

We can express \( b \) and \( c \) in terms of \( \alpha \) and \( \beta \):
- \( b = -a(\alpha + \beta) \)
- \( c = a\alpha\beta \)

7. **Substituting back into the formulas**:
Substitute these values into the expressions for \( P + Q \) and \( PQ \):
- \( P + Q = -\frac{-a(\alpha + \beta) - 2a\alpha\beta}{a - (-a(\alpha + \beta)) + a\alpha\beta} \)
- \( PQ = \frac{a\alpha\beta}{a - (-a(\alpha + \beta)) + a\alpha\beta} \)

8. **Simplifying the expressions**:
After simplification, we find:
- The roots of the new equation are:
\[ P = \frac{\alpha}{1 + \alpha}, \quad Q = \frac{\beta}{1 + \beta} \]

### Conclusion:
Thus, the roots of the equation \( ax^2 - bx(x-1) + c(x-1)^2 = 0 \) are \( \frac{\alpha}{1 + \alpha} \) and \( \frac{\beta}{1 + \beta} \).