Multiple choice

For how many integral values of k, the equation x2 - 4x + k = 0, where k is an integer, has real roots and both of them lie in the interval (0, 5)?

  1. 3

  2. 4

  3. 5

  4. 6
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For roots to lie in (0, 5), the discriminant D = 16 - 4k must be >= 0 (so k <= 4). Also, f(0) > 0 (k > 0), f(5) > 0 (25 - 20 + k > 0 => k > -5), and the vertex 2 must be in (0, 5). Thus, k must be 1, 2, 3, 4.

AI explanation

The given equation is (x - 2)2 + k - 4 = 0, so the roots are 2 + sqrt(4 - k) and 2 - sqrt(4 - k). For the roots to be real, the discriminant (4 - k) must be non-negative, meaning k <= 4. For both roots to lie in the interval (0, 5), the smaller root 2 - sqrt(4 - k) must be greater than 0. This means sqrt(4 - k) < 2, which leads to 4 - k < 4 and thus k > 0. The integer values of k satisfying 0 < k <= 4 are 1, 2, 3 and 4, giving 4 values.

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