Quadratic Equations and Inequalities: Integer roots MCQ

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Integer rootsHardQuestion 220015

Question

The number of integer values of k for which x² − 2(k−1)x + 4 > 0 for all real x is:

A
1
B
2
C
3
D
Infinite
Correct

Correct answer

Infinite

Explanation

For the quadratic to be > 0 for all x, we need a > 0 (✓, a=1) and D < 0. D = 4(k−1)² − 16 < 0 → (k−1)² < 4 → −2 < k−1 < 2 → −1 < k < 3. Integer values: k ∈ {0, 1, 2} — three values, but the question may allow: actually k = 0, 1, 2 are 3 integer values.

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