If a^3 - b^3 = 56 and a - b = 2 for real numbers a and b, then what is the value of a^2 + b^2?

Introduction / Context:
This is an algebra question that uses standard identities involving sums and differences of powers. You are given values for a^3 - b^3 and a - b and asked to determine a^2 + b^2. Such questions test your familiarity with algebraic formulas and your ability to manipulate them to find the required expression without necessarily solving for a and b individually.

Given Data / Assumptions:

a and b are real numbers.
a^3 - b^3 = 56 is given.
a - b = 2 is also given.
We need to find the numerical value of a^2 + b^2.
We assume basic algebraic identities hold for a and b.

Concept / Approach:
The key idea is to use the factorisation identity for the difference of cubes and the identity for the square of a difference. The difference of cubes factorisation is a^3 - b^3 = (a - b) * (a^2 + ab + b^2). We already know a - b and a^3 - b^3, so we can find the value of a^2 + ab + b^2. Then we use the identity (a - b)^2 = a^2 + b^2 - 2ab, together with the previous result, to solve for a^2 + b^2 and ab. This method is efficient and avoids solving the full system for a and b.

Step-by-Step Solution:
Step 1: Use the identity for the difference of cubes: a^3 - b^3 = (a - b) * (a^2 + ab + b^2). Step 2: Substitute the given values: 56 = (a - b) * (a^2 + ab + b^2) and a - b = 2. Step 3: Replace a - b with 2 to get 56 = 2 * (a^2 + ab + b^2). Step 4: Divide both sides by 2 to obtain a^2 + ab + b^2 = 28. Step 5: Next, use the identity for the square of a difference: (a - b)^2 = a^2 + b^2 - 2ab. Step 6: Substitute a - b = 2, so (a - b)^2 = 2^2 = 4, and thus a^2 + b^2 - 2ab = 4. Step 7: Let S = a^2 + b^2 and P = ab. Then from Step 4 we have S + P = 28, and from Step 6 we have S - 2P = 4. Step 8: Solve this system of two linear equations in S and P. Subtract the second equation from the first: (S + P) - (S - 2P) = 28 - 4. Step 9: This gives 3P = 24, so P = 24 / 3 = 8. Step 10: Substitute P = 8 back into S + P = 28 to obtain S + 8 = 28, so S = 20. Step 11: Therefore a^2 + b^2 = S = 20.

Verification / Alternative check:
An alternative check is to look for integer solutions that satisfy both given conditions. Suppose a and b are integers with a - b = 2. Try b = 2, then a = 4. Compute a^3 - b^3 = 4^3 - 2^3 = 64 - 8 = 56, which matches the given condition. Now compute a^2 + b^2 = 4^2 + 2^2 = 16 + 4 = 20. This matches the algebraic result obtained using identities, confirming that a^2 + b^2 equals 20.

Why Other Options Are Wrong:
12: This value would imply different relationships between a and b and does not satisfy both given equations, so it is inconsistent with the data.
28: This is the value of a^2 + ab + b^2, not of a^2 + b^2, so it represents a misreading or incomplete use of the identity.
32: This value would require a and b to satisfy different numerical conditions and does not match the correct system solution derived from the identities.

Common Pitfalls:
Students often stop after finding a^2 + ab + b^2 = 28 and mistakenly mark 28 as the answer, confusing it with a^2 + b^2. Another common mistake is to try to guess a and b without using identities, which can be time consuming if the numbers are not obvious. Some candidates also misuse the identity for (a + b)^2 instead of (a - b)^2, which leads to incorrect equations. To avoid these errors, always read the exact expression asked for in the question and apply the correct algebraic identity step by step.

Final Answer:
The value of a^2 + b^2 is 20.