The sum of the squares of two numbers is 234, and the square of their difference is 144. Find the product of the two numbers.

Introduction / Context:This problem uses algebraic identities that relate sums and differences to products. Specifically, knowing a^2 + b^2 and (a − b)^2 enables solving for ab via identity manipulations without finding a and b individually.

Given Data / Assumptions:

• a^2 + b^2 = 234.
• (a − b)^2 = 144.
• Need ab.

Concept / Approach:Use the identity (a − b)^2 = a^2 + b^2 − 2ab. Rearranging gives 2ab = (a^2 + b^2) − (a − b)^2. Substitute the given values to compute ab directly.

Step-by-Step Solution:

Verification / Alternative check:If desired, we can choose numbers with product 45 that fit the constraints (not necessary here). The identity-based computation is sufficient and exact.

Why Other Options Are Wrong:

• 28, 52, 36, 40: These do not satisfy 2ab = 90; substituting back fails the identity relation.

Common Pitfalls:

• Using (a + b)^2 identity mistakenly instead of (a − b)^2.
• Sign errors when isolating 2ab.

Final Answer:45