If a^3 + b^3 = 19 and a + b = 1 for real numbers a and b, what is the value of the product ab obtained using the standard cubic identity for the sum of two cubes?

Introduction / Context:
This algebra question focuses on using the identity for the sum of cubes to connect the sum a + b, the sum of cubes a^3 + b^3, and the product ab. Such identities are very useful in aptitude problems where direct factorisation may not be obvious but standard formulas apply neatly.

Given Data / Assumptions:

• a^3 + b^3 = 19.
• a + b = 1.
• a and b are real numbers.
• We must find ab.

Concept / Approach:
The standard identity for the sum of cubes is a^3 + b^3 = (a + b)^3 − 3ab(a + b). We are already given a^3 + b^3 and a + b, so we can substitute these values into the identity and solve for ab. This transforms the problem into solving a simple linear equation in ab.

Step-by-Step Solution:
Use the identity a^3 + b^3 = (a + b)^3 − 3ab(a + b). Given a^3 + b^3 = 19 and a + b = 1. Substitute into the identity: 19 = (1)^3 − 3ab(1). So 19 = 1 − 3ab. Rearrange to isolate ab: 1 − 3ab = 19. Thus −3ab = 19 − 1 = 18. So ab = −18 / 3 = −6.
Verification / Alternative check:
As a quick consistency check, we could attempt to find specific values of a and b that satisfy a + b = 1 and ab = −6 by solving the quadratic t^2 − (a + b)t + ab = 0, that is t^2 − t − 6 = 0. This quadratic has roots t = 3 and t = −2. For a = 3 and b = −2, indeed a + b = 1 and a^3 + b^3 = 27 − 8 = 19, confirming that ab = −6 works.

Why Other Options Are Wrong:
Options a, c, d, and e do not satisfy the identity when substituted. For example, if ab were 5 or 7, the expression (a + b)^3 − 3ab(a + b) would not equal 19 with a + b = 1. Only ab = −6 maintains the given relationship between the sum of cubes and the sum of the numbers.

Common Pitfalls:
Some learners mistakenly use the difference of cubes identity or try to factor a^3 + b^3 directly as (a + b)(a^2 − ab + b^2) without connecting it to the known value of a + b. While that path can also work, it usually leads to more algebra. Using the cubic identity with the known sum is the most efficient approach here.

Final Answer:
The required product of the two numbers is ab = −6.