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Consider the set S comprising all non-zero real values of α such that the quadratic equation αx² - x + α = 0 possesses two distinct real roots x₁ and x₂, which satisfy the condition |x₁ - x₂| < 1. Which of the following intervals lies entirely within S?

1. (A) (-1/2, -1/√5)
2. (B) (-1/√5, 0)
3. (C) (0, 1/√5)
4. (D) (1/√5, 1/2)

Correct answer: (A) (-1/2, -1/√5)

Solution

The condition for two distinct real roots with |x₁ - x₂| < 1 is derived from the discriminant and root separation formula. The interval (-1/2, -1/√5) satisfies this condition entirely.

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