The number of integral values of k such that both distinct roots of the quadratic equation x^2 - 2kx + k^2 - k = 0 lie strictly inside the interval (1, 7) is :

Step-by-step solution

Let f(x) = x^2 - 2kx + k^2 - k . For both distinct roots to lie strictly inside the interval (1, 7) , the following four conditions must be satisfied: 1. Discriminant D > 0 4k^2 - 4(k^2 - k) > 0 4k > 0 k > 0 2. Vertex x-coordinate lies in (1, 7) 1 3. f(1) > 0 1 - 2k + k^2 - k > 0 k^2 - 3k + 1 > 0 The roots are 3 5 2 . Thus, k 3 + 5 2 (approx 2.62 ). 4. f(7) > 0 49 - 14k + k^2 - k > 0 k^2 - 15k + 49 > 0 The roots are 15 29 2 . Thus, k 15 + 29 2 (approx 10.19 ). Taking the intersection of all these conditions: From (