If a − b = 1 and a³ − b³ = 61 for two real numbers a and b, what is the exact value of the product ab?

Introduction / Context:
This problem uses standard algebraic identities involving cubes and differences of numbers. By combining the identity for a³ − b³ with the given difference a − b, we can find the product ab without solving a full quadratic. Such questions test your familiarity with identities and your ability to manipulate them efficiently.

Given Data / Assumptions:

• a − b = 1.
• a³ − b³ = 61.
• a and b are real numbers.
• We are required to find ab.

Concept / Approach:
Key ideas:

• Use the identity a³ − b³ = (a − b)(a² + ab + b²).
• Use the identity (a − b)² = a² + b² − 2ab to connect a² + b² with ab.
• First find a² + ab + b² from the cubic identity, then express a² + b² in terms of ab using the squared difference and finally solve for ab.

Step-by-Step Solution:
Step 1: Apply the identity a³ − b³ = (a − b)(a² + ab + b²). Step 2: Substitute the known values: 61 = (a − b)(a² + ab + b²) = 1 × (a² + ab + b²). Step 3: So a² + ab + b² = 61. Step 4: Now use (a − b)² = a² + b² − 2ab. Since a − b = 1, we have 1² = a² + b² − 2ab, so a² + b² = 1 + 2ab. Step 5: Substitute this into a² + ab + b² = 61 to get (1 + 2ab) + ab = 61. Step 6: Combine like terms: 1 + 3ab = 61. Step 7: Rearrange to find ab: 3ab = 60, hence ab = 60 / 3 = 20.
Verification / Alternative check:
Step 1: Once ab = 20 is found, we can construct the quadratic whose roots are a and b with sum a + b and product ab, although the exact roots are not necessary. Step 2: From a − b = 1 and ab = 20, one can solve for a and b explicitly and then compute a³ − b³ numerically to confirm that it equals 61. Step 3: This backwards check, while more tedious, would verify the consistency of the result.
Why Other Options Are Wrong:
Option −20 would give a² + b² = 1 + 2(−20) = −39, which is not compatible with a² + ab + b² = 61. Option 30 would lead to a² + b² = 1 + 60 = 61, making a² + ab + b² = 61 + 30 = 91, not 61. Option 60 gives a² + b² = 121 and then a² + ab + b² = 181, again contradicting the given cubic equation value. Option 10 similarly fails to satisfy the identity when substituted back.
Common Pitfalls:
A common error is to try to expand a³ − b³ directly without using the identity, which becomes messy. Some students incorrectly apply the square or cube identities or mix up a³ + b³ with a³ − b³. Forgetting the relationship between a² + b² and ab via (a − b)² leads to unnecessary algebra or incorrect assumptions.
Final Answer:
The value of the product ab is 20.