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For the quadratic equation x² - 2kx + k² + k - 5 = 0, if each of its two roots is smaller than 5, then the parameter k must belong to which interval?

1. (5, 6]
2. (6, ∞)
3. (-∞, 4)
4. [4, 5]

Correct answer: (-∞, 4)

Solution

The roots of the quadratic equation must be less than 5, which can be determined by analyzing the vertex and the behavior of the parabola. For the roots to be smaller than 5, the value of k must be such that the vertex of the parabola, which occurs at x = k, is less than 5, leading to the conclusion that k must be in the interval (-∞, 4).

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