The coefficients a, b and c of the quadratic equation, ax² + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :

Solution

Maths sir · 3 steps

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

1. Condition and total. Equal roots need the discriminant to be zero, so b² = 4ac. Each of a, b, c is a die score from 1 to 6, so there are 6 × 6 × 6 = 216 equally likely outcomes.
2. Only even b works. Since 4ac is a multiple of 4, b² must be too. So b is even: 2, 4 or 6. For each one, ac = b²/4.

   b	ac = b²/4	Ordered pairs (a, c)	Count
   1, 3, 5	not a whole number	none	0
   2	1	(1, 1)	1
   4	4	(1, 4), (4, 1), (2, 2)	3
   6	9	(3, 3)	1

3. So. There are 1 + 3 + 1 = 5 favourable outcomes, so the probability is 5/216, which is option (b).

The idea to keep. Turn the root condition into an equation in the die scores, then fix one variable and count the ordered cases. Each throw is a separate event, so (1, 4) and (4, 1) count as two.

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