The coefficients a, b, c in the quadratic equation a x²+b x+c=0 are chosen from the set {1,2,3,4,5,6,7,8}. The probability of this equation having repeated roots is :

Solution

Maths sir · 3 steps

Repeated roots means b² = 4ac, so fix b and count the ordered pairs (a, c) that work.

1. Count all equations. Each of a, b, c can be any of the 8 numbers, and repeats are allowed. That gives 8 × 8 × 8 = 512 equally likely equations.
2. Condition for repeated roots. The discriminant must be zero, so b² = 4ac. That makes b² a multiple of 4, so b must be even: b = 2, 4, 6 or 8. For each one, ac = b²/4.

   b	ac = b²/4	Ordered pairs (a, c) from 1 to 8	Count
   2	1	(1,1)	1
   4	4	(1,4), (4,1), (2,2)	3
   6	9	(3,3) only, since 9 is not in the set	1
   8	16	(2,8), (8,2), (4,4)	3

3. So. There are 1 + 3 + 1 + 3 = 8 favourable equations. The probability is 8/512 = 1/64, which is option (b).

The idea to keep. For a discriminant condition, fix the middle coefficient and count ordered pairs (a, c). Here a and c play different roles, so (1,4) and (4,1) are two different equations.

Answer from the NTA final answer key. Solution by PrepWiser.