The sum of squares of two numbers is 68, and the square of their difference is 36. Find the product of the two numbers.

Introduction / Context:This question leverages the identity for the square of a difference and its relationship to the product term. With aggregate information (sum of squares and squared difference), we can deduce the product directly without solving for individual numbers.

Given Data / Assumptions:

• a^2 + b^2 = 68.
• (a − b)^2 = 36.
• We need ab.

Concept / Approach:Use the identity (a − b)^2 = a^2 + b^2 − 2ab. Plug in the given values and solve for ab. This bypasses solving a quadratic for a and b and goes straight to the required product.

Step-by-Step Solution:

Verification / Alternative check:Optional: (a + b)^2 = a^2 + b^2 + 2ab = 68 + 32 = 100 → a + b = 10 is consistent with real solutions (e.g., numbers satisfying sum 10 and product 16 exist).

Why Other Options Are Wrong:32 represents 2ab, not ab. 58 and 104 are inconsistent with the identity. 22 is arbitrary here.

Common Pitfalls:Forgetting to divide by 2 after finding 2ab, or mixing identities for (a − b)^2 and (a + b)^2.

Final Answer:16