Use algebraic identities effectively: If a^3 − b^3 = 56 and a − b = 2, find the value of (a^2 + b^2).

Introduction / Context:This problem is a classic application of the factorization identity for cubes together with simple system solving. It checks whether you can relate a^3 − b^3 and a − b to obtain expressions for a^2 + b^2 and ab.

Given Data / Assumptions:

• a^3 − b^3 = 56.
• a − b = 2.
• a and b are real numbers.

Concept / Approach:Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2) to find a^2 + ab + b^2. Then use (a − b)^2 = a^2 + b^2 − 2ab to form two equations in the unknowns S = a^2 + b^2 and P = ab, and solve for S.

Step-by-Step Solution:

Verification / Alternative check:Choose numbers satisfying a − b = 2 and ab = 8, e.g., solve t^2 − 2t + 8 = 0 to get complex roots that still satisfy the identities; S remains 20 by construction, so the value is consistent with the algebraic relations.

Why Other Options Are Wrong:18, −10, −12, and 16 arise from arithmetic slips when solving for P or mixing signs in the square identity.

Common Pitfalls:Mistaking a^2 + b^2 + ab for a^2 + b^2, or using (a − b)^2 incorrectly as a^2 + b^2 + 2ab rather than a^2 + b^2 − 2ab.

Final Answer:20