The coefficients a, b, c of the quadratic equation a x^2+b x+c=0 are chosen independently with replacement from the set 1,2,3,4,5,6 . The probability that this equation has imaginary roots is :

Step-by-step solution

Total number of possible equations is 6 6 6 = 216 . For the quadratic equation a x^2+b x+c=0 to have imaginary roots, the discriminant must be strictly less than zero: D = b^2 - 4ac It is easier to count the complementary event, which is the equation having real roots ( b^2 4ac ). Let us find the number of ordered triplets (a, b, c) satisfying this condition: If b = 1 , 1 4ac no valid pairs since a, c 1 . If b = 2 , 4 4ac ac 1 (1, 1) (1 case). If b = 3 , 9 4ac ac 2 (1, 1), (1, 2), (2, 1) (3 cases). If b = 4 , 16 4a