If A and B are the roots of the equation ` A x^(2) - A^(2) x + AB = 0` , then what are the value of A and B respectivley ?

Options

A. `1, 0`
B. `1, 1`
C. `0, 2`
D. `0, 1`

Answer

To solve the equation \( A x^2 - A^2 x + AB = 0 \) and find the values of \( A \) and \( B \) respectively, we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = A \)
- \( b = -A^2 \)
- \( c = AB \)

### Step 2: Use the sum and product of roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots (let's denote them as \( A \) and \( B \)) is given by:
\[ A + B = -\frac{b}{a} \]
Substituting the values of \( a \) and \( b \):
\[ A + B = -\frac{-A^2}{A} = \frac{A^2}{A} = A \]

### Step 3: Calculate the product of the roots
The product of the roots is given by:
\[ AB = \frac{c}{a} \]
Substituting the values of \( c \) and \( a \):
\[ AB = \frac{AB}{A} \]
This simplifies to:
\[ AB = B \]

### Step 4: Solve the equations
From the sum of the roots, we have:
1. \( A + B = A \)
2. From the product of the roots, we have:
- If \( AB = B \), we can rearrange this to \( AB - B = 0 \) or \( B(A - 1) = 0 \).

This gives us two cases:
- Case 1: \( B = 0 \)
- Case 2: \( A - 1 = 0 \) which gives \( A = 1 \)

### Step 5: Find the values of \( A \) and \( B \)
From Case 1, if \( B = 0 \):
- Substitute \( B = 0 \) into \( A + B = A \):
\[ A + 0 = A \quad \text{(which is true)} \]
Thus, we can conclude that:
- \( A = 1 \)
- \( B = 0 \)

### Final Answer
The values of \( A \) and \( B \) respectively are:
\[ A = 1, \quad B = 0 \]