Nature of roots for x^2 − x − 2 = 0: Which statement is correct about its two roots?

Introduction / Context:We analyze the quadratic x^2 − x − 2 = 0 to determine qualitative properties of its roots. Factoring cleanly reveals exact integer solutions, allowing us to evaluate each statement provided.Given Data / Assumptions:

• Equation: x^2 − x − 2 = 0.

Concept / Approach:Factor to find roots. Then check whether both are integers, both natural numbers, or whether a specific sign statement holds.Step-by-Step Solution:

Verification / Alternative check:Plug back: 2^2 − 2 − 2 = 0 and (−1)^2 − (−1) − 2 = 1 + 1 − 2 = 0.

Why Other Options Are Wrong:

• both of them are natural numbers: False since −1 is not natural.
• the latter of the two is negative: Ambiguous phrasing; the set has one negative and one positive. A clearer property is “one root is negative and one is positive.”
• None of these: Incorrect because “both integers” is true.

Common Pitfalls:Depending on definitions of “natural numbers.” Most exams use 1,2,3,…, thus excluding −1 and 0.

Final Answer: