Two real numbers a and b satisfy: the sum of their squares is 97 and the square of their difference is 25. Using these relations, determine the product a * b.

Introduction / Context:
This quantitative aptitude question tests algebraic identities that connect sums/differences of numbers with their products. Instead of finding the numbers explicitly, you can use standard formulas to extract the product directly from aggregate information like a^2 + b^2 and (a − b)^2.

Given Data / Assumptions:

• a^2 + b^2 = 97.
• (a − b)^2 = 25.
• a and b are real numbers; the task is to compute a * b.

Concept / Approach:
The key identity is (a − b)^2 = a^2 + b^2 − 2ab. Since a^2 + b^2 and (a − b)^2 are known, you can isolate 2ab and then divide by 2 to get the product ab. This avoids solving for a and b individually and minimizes arithmetic steps.

Step-by-Step Solution:

Verification / Alternative check:
As a consistency check, compute (a + b)^2 = a^2 + b^2 + 2ab = 97 + 72 = 169, so a + b = 13 or −13. Real number pairs with sum 13 and product 36 do exist (they are roots of t^2 − 13t + 36 = 0), confirming that the data are internally consistent.

Why Other Options Are Wrong:
45, 54, and 63 arise from misplacing terms in the identity or using (a + b)^2 incorrectly. 28 is an arbitrary distractor that does not satisfy 2ab = 72.

Common Pitfalls:
Confusing (a − b)^2 with (a + b)^2, or forgetting to divide 2ab by 2 to get ab. Another mistake is attempting to compute a and b explicitly, which is unnecessary here.

Final Answer:
36