If a and b are real numbers such that a + b = 1 and ab = -6, find the exact value of a^3 + b^3. Use standard algebraic identities (do not approximate), and keep the given values a + b = 1 and ab = -6 unchanged.

Introduction / Context: This question tests the identity that relates the sum of cubes (a^3 + b^3) to the sum (a + b) and product (ab). Instead of solving for a and b individually, we can use an algebraic shortcut that works for all real numbers where the given conditions hold. This is a common simplification pattern in aptitude: convert a complicated-looking expression into something you can compute directly from provided symmetric values.

Given Data / Assumptions:

• a + b = 1 • ab = -6 • a and b are real numbers • Required: a^3 + b^3

Concept / Approach: Use the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b). This identity is derived from expanding (a + b)^3 and regrouping terms. It is especially useful when a + b and ab are given directly, because it avoids solving a quadratic for a and b.

Step-by-Step Solution: 1) Start with the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b) 2) Substitute a + b = 1: (a + b)^3 = 1^3 = 1 3) Substitute ab = -6 and a + b = 1 into the second term: -3ab(a + b) = -3 * (-6) * 1 = 18 4) Add the results: a^3 + b^3 = 1 + 18 = 19

Verification / Alternative check: If you form the quadratic with roots a and b: t^2 - (a + b)t + ab = 0 gives t^2 - t - 6 = 0, which factors to (t - 3)(t + 2) = 0. So (a, b) could be (3, -2) in some order. Then a^3 + b^3 = 27 + (-8) = 19, confirming the identity-based result.

Why Other Options Are Wrong: • 17, 15, 13, 21: these do not match the identity result using the fixed values a + b = 1 and ab = -6.

Common Pitfalls: • Using a^3 + b^3 = (a + b)^3 + 3ab(a + b) with the wrong sign. • Trying to guess a and b without using the symmetric identity.

Final Answer: 19