The coefficients a, b and c of the quadratic equation, a x²+b x+c=0 are obtained by throwing a dice three times. The probability that this equation has equal roots is

Step-by-step solution

The quadratic equation ax² + bx + c = 0 has equal roots if the discriminant D = b² - 4ac = 0 , which implies b² = 4ac . Since a, b, c are outcomes of throwing a dice three times, a, b, c 1, 2, 3, 4, 5, 6 . The total number of possible outcomes is 6 6 6 = 216 . We need to find the number of triplets (a, b, c) such that b² = 4ac . If b = 1 , b² = 1 , 4ac = 1 (no integer solution). If b = 2 , b² = 4 , 4ac = 4 ac = 1 . The only solution is (a, c) = (1, 1) . This gives 1 triplet: (1, 2, 1) . If b = 3 , b² = 9 , 4ac = 9