In basic algebra, if a − b = −1 and the product ab = 6, then what is the exact value of the difference of cubes a^3 − b^3?

Introduction / Context:
This algebra question tests the use of standard identities for the difference of cubes and for the square of a binomial. Instead of solving directly for a and b, we can work with expressions such as a − b, ab, and a^2 + b^2 to reach the required expression a^3 − b^3. This technique is common in aptitude exams where the goal is to manipulate expressions efficiently without solving a full quadratic equation.

Given Data / Assumptions:

• a − b = −1
• ab = 6
• a and b are real or complex numbers, but the identities hold for all algebraic numbers
• We need to find a^3 − b^3

Concept / Approach:
We use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). We already know a − b and ab. To find a^2 + b^2 we use the binomial square identity (a − b)^2 = a^2 + b^2 − 2ab. Once we compute a^2 + b^2, we can add ab to get a^2 + ab + b^2 and then multiply by a − b to get the required difference of cubes.

Step-by-Step Solution:
Start from (a − b)^2 = a^2 + b^2 − 2ab. Given a − b = −1, so (a − b)^2 = (−1)^2 = 1. Substitute the known values: 1 = a^2 + b^2 − 2ab. Use ab = 6, so −2ab = −2 * 6 = −12. Therefore 1 = a^2 + b^2 − 12, so a^2 + b^2 = 13. Now find a^2 + ab + b^2 = (a^2 + b^2) + ab = 13 + 6 = 19. Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). Substitute a − b = −1 and a^2 + ab + b^2 = 19. Then a^3 − b^3 = (−1) * 19 = −19.

Verification / Alternative check:
We can verify indirectly by constructing a quadratic with roots a and b. The sum a + b would satisfy t^2 − (a + b)t + ab = 0. From a − b and ab there is no need to solve fully because the identities used are exact. The internal consistency of a^2 + b^2 = 13 and ab = 6 leads to a^2 + ab + b^2 = 19, which correctly reproduces the result −19 when multiplied by a − b. This confirms the algebraic steps are valid and that no arithmetic mistake occurred in the process.

Why Other Options Are Wrong:
33 and 35 arise if one misuses signs in the binomial square identity or in the final multiplication. 18 appears if ab is used incorrectly in forming a^2 + ab + b^2. The value 0 would require a^3 = b^3, which is impossible when a − b = −1. Hence these values do not match the correct algebraic manipulation.

Common Pitfalls:
A common mistake is to confuse (a − b)^2 with a^2 − b^2, which is a different identity. Another frequent error is to forget the factor 2ab when expanding the square or to substitute ab with the wrong sign. Some learners also attempt to find explicit values of a and b even though it is unnecessary and may lead to messy quadratic equations. Working symbolically with identities is more efficient and reduces computational errors in timed exams.

Final Answer:
Thus, the exact value of the difference of cubes is −19.