if `alpha&beta`are the roots of the quadratic equation `ax^2 + bx + c = 0`, then the quadratic equation `ax^2-bx(x-1)+c(x-1)^2 =0` has roots

Options

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

Answer

To solve the problem, we need to find the roots of the quadratic equation given by:

\[ ax^2 - bx(x - 1) + c(x - 1)^2 = 0 \]

### Step 1: Expand the equation
Start by expanding the terms in the equation:

\[ ax^2 - bx(x - 1) + c(x - 1)^2 = ax^2 - bx^2 + bx + c(x^2 - 2x + 1) \]

Expanding \(c(x - 1)^2\):

\[ c(x - 1)^2 = c(x^2 - 2x + 1) = cx^2 - 2cx + c \]

Now, substituting this back into the equation:

\[ ax^2 - bx^2 + bx + cx^2 - 2cx + c = 0 \]

### Step 2: Combine like terms
Now, combine the \(x^2\) terms and the \(x\) terms:

\[ (a - b + c)x^2 + (b - 2c)x + c = 0 \]

### Step 3: Identify coefficients
From the combined equation, we can identify the coefficients:

- Coefficient of \(x^2\): \(A = a - b + c\)
- Coefficient of \(x\): \(B = b - 2c\)
- Constant term: \(C = c\)

### Step 4: Find the sum and product of the roots
Using Vieta's formulas, we know:

1. The sum of the roots \(P + Q = -\frac{B}{A} = -\frac{b - 2c}{a - b + c}\)
2. The product of the roots \(PQ = \frac{C}{A} = \frac{c}{a - b + c}\)

### Step 5: Substitute known values
We know from the original quadratic equation \(ax^2 + bx + c = 0\) that:

- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

### Step 6: Express \(P + Q\) and \(PQ\) in terms of \(\alpha\) and \(\beta\)
Now, we can express \(P + Q\) and \(PQ\) in terms of \(\alpha\) and \(\beta\):

1. For the sum:
\[ P + Q = -\frac{b - 2c}{a - b + c} = \frac{2c - b}{a - b + c} \]
Substituting \(b = -a(\alpha + \beta)\) and \(c = a\alpha\beta\), we can express this in terms of \(\alpha\) and \(\beta\).

2. For the product:
\[ PQ = \frac{c}{a - b + c} = \frac{c}{a - (-a(\alpha + \beta)) + c} = \frac{c}{a(1 + \alpha + \beta) + c} \]

### Final Step: Roots in terms of \(\alpha\) and \(\beta\)
After simplification, we find that:

\[ P = \frac{\alpha}{1 + \alpha}, \quad Q = \frac{\beta}{1 + \beta} \]

Thus, the roots of the quadratic equation \(ax^2 - bx(x - 1) + c(x - 1)^2 = 0\) are:

\[ P = \frac{\alpha}{1 + \alpha}, \quad Q = \frac{\beta}{1 + \beta} \]