The sum of squares of two numbers is 80 and the square of their difference is 36. What is the product of the two numbers?

Introduction / Context:
This question rewards familiarity with the identity for the square of a difference and its relationship to the product of two numbers. With two aggregate pieces of information—sum of squares and square of difference—you can extract the product directly without finding the individual numbers first.

Given Data / Assumptions:

• a^2 + b^2 = 80.
• (a − b)^2 = 36.
• We need ab.

Concept / Approach:
Use (a − b)^2 = a^2 + b^2 − 2ab. Since both a^2 + b^2 and (a − b)^2 are known, rearrange to solve for ab. This approach avoids solving any quadratic for a and b explicitly.

Step-by-Step Solution:

Verification / Alternative check:
If needed, note that (a + b)^2 = a^2 + b^2 + 2ab = 80 + 44 = 124, so a + b = sqrt(124). This is consistent with real values for a and b; no contradiction arises.

Why Other Options Are Wrong:
44 is 2ab, not ab. Values 58 and 116 do not align with the given identities. 18 is arbitrary here.

Common Pitfalls:
Stopping at 2ab = 44 and reporting 44 instead of dividing by 2; mixing up the identities for (a − b)^2 and (a + b)^2.

Final Answer:
22