Multiple choice

If the equation x2 - 2x + |k| = 0, where k belongs to integers, has two real and distinct roots, then the values of k lie in the interval ____.

  1. (-2, 2)

  2. (-2, 1]

  3. (-1, 1)

  4. (-1, 2]

  5. (-2, 1)
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C Correct answer
Explanation

For the quadratic equation to have two real and distinct roots, its discriminant must be strictly greater than zero. This gives 4 - 4|k| > 0, which simplifies to |k| < 1, meaning k must lie in the open interval (-1, 1). Since k is specified to be an integer, the only possible value for k is 0, which lies within this interval.

AI explanation

For the quadratic equation x^2 - 2x + |k| = 0 to have two real and distinct roots, the discriminant must be strictly greater than zero. The discriminant is (-2)^2 - 4(1)(|k|), which simplifies to 4 - 4|k|. Setting this greater than zero gives 4 - 4|k| > 0, so |k| < 1. This inequality means k lies strictly between -1 and 1.

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