Consider the quadratic equation A x^2 − A^2 x + A B = 0, where A and B are real constants and A ≠ 0. If the roots of this equation are exactly A and B, then what are the values of A and B respectively?

Introduction / Context:
This question mixes parameter coefficients and roots using the same letters A and B, which can be slightly confusing. The idea is that A and B appear both as constants in the quadratic equation and as the actual roots of the equation. We must determine numerical values of A and B that make this self consistent.

Given Data / Assumptions:

• Quadratic equation: A x^2 − A^2 x + A B = 0.
• A and B are real constants.
• A ≠ 0 so that the equation is genuinely quadratic.
• The roots of this equation are exactly A and B.
• We must identify the correct ordered pair (A, B) from the options.

Concept / Approach:
If the roots of the equation are A and B, then they must satisfy the standard root relationships. For any quadratic k x^2 + l x + m = 0 with roots r1 and r2, we have r1 + r2 = −l / k and r1 r2 = m / k. In our case k = A, l = −A^2, and m = A B. We can express the sum and product of roots first in terms of A and B via these formulas, and also from the condition that the roots themselves are A and B. Equating the two sets of expressions allows us to solve for A and B.

Step-by-Step Solution:
1. Identify coefficients: in the equation A x^2 − A^2 x + A B = 0, k = A, l = −A^2, and m = A B. 2. According to the root relationships, sum of roots = −l / k = −(−A^2) / A = A. 3. Product of roots = m / k = A B / A = B. 4. By the condition in the question, the roots themselves are A and B. 5. Therefore, sum of roots (from the roots directly) is A + B. 6. We equate this to the coefficient based sum: A + B = A. 7. This simplifies to B = 0. 8. For the product, from the roots directly we have A * B. 9. From the coefficients we know product = B. 10. So A * B must equal B. 11. Substituting B = 0 gives A * 0 = 0, which equals B, so this condition is satisfied for any A. 12. However, for the equation to be quadratic we require A ≠ 0. 13. Among the answer choices, only the pair A = 1, B = 0 has A non zero and B equal to 0.

Verification / Alternative check:
Substitute A = 1 and B = 0 into the original equation. Then the equation becomes 1 x^2 − 1^2 x + 1 * 0 = x^2 − x = 0. Its roots are 0 and 1. These correspond exactly to B = 0 and A = 1, confirming that the choice A = 1, B = 0 works. Any option with A = 0 would make the x^2 coefficient zero, so the equation would not be quadratic, contradicting the setup of the problem.

Why Other Options Are Wrong:
Option B (A = 1, B = 1) would make the equation x^2 − x + 1 = 0. The roots of this equation are complex and not equal to 1 and 1, so this option fails.
Options C and D both have A = 0. In these cases, the term A x^2 disappears, and the equation is no longer quadratic, violating the condition that A is the leading coefficient of a quadratic equation.

Common Pitfalls:
A major pitfall here is confusing the role of the letter A as both coefficient and root. Some learners try to treat A and B as arbitrary parameters without imposing the root condition. Others forget to enforce that A must be non zero for the equation to be genuinely quadratic. Carefully distinguishing between coefficients and roots while using the standard sum and product formulas helps avoid these mistakes.

Final Answer:
The correct values are A = 1, B = 0.