Let a, b, c 1, 2, 3, 4 . If the probability, that ax^2 + 2 2 ,bx + c > 0 for all x R , is m n , (m, n) = 1 , then m + n is equal to _______.

Step-by-step solution

For the quadratic expression ax^2 + 2 2 bx + c > 0 for all x R , the leading coefficient must be positive and the discriminant must be negative. Since a 1, 2, 3, 4 , the condition a > 0 is always satisfied. The discriminant condition is: D (2 2 b)^2 - 4ac 8b^2 - 4ac 2b^2 The total number of possible triplets (a, b, c) is 4 4 4 = 64 . Now, we find the number of favorable outcomes by checking the possible values of b 1, 2, 3, 4 : Case 1: b = 1 We need 2(1)^2 2 . The total number of pairs (a, c) is 4 4 = 16 . The pair