Nature of roots for x^2 − x − 2 = 0: Which statement is correct about its two roots?
Aptitude
Quadratic Equation
Difficulty: Easy
Choose an option
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Aboth of them are natural numbers
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Bboth of them are integers
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Cthe latter of the two is negative
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DNone of these
Answer
Correct Answer: both of them are integers
Explanation
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:
x^2 − x − 2 = (x − 2)(x + 1) = 0.Roots are x = 2 and x = −1.Both are integers; only one of them (2) is a natural number if we adopt the common convention that natural numbers are positive integers starting from 1.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:
both of them are integers