Explanation
1. Condition for Equal Roots
A quadratic equation ax2+bx+c=0 has equal roots if and only if its discriminant (D) is zero:
2. Total Outcomes
Since a,b, and c are obtained by rolling a die thrice, each can take values from {1,2,3,4,5,6}.
3. Favorable Outcomes
We need to find the number of triples (a,b,c) such that b2=4ac. Note that for b2 to be equal to 4ac, b2 must be a multiple of 4, which means b must be an even number (b∈{2,4,6}).
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Case 1: b=2
The only possibility is (a,c)=(1,1).
Triplet: (1,2,1) (1 case)
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Case 2: b=4
Possible pairs for (a,c) are (1,4),(2,2), and (4,1).
Triplets: (1,4,4),(2,4,2),(4,4,1) (3 cases)
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Case 3: b=6
The only possibility within the die range {1,…,6} is (a,c)=(3,3).
Triplet: (3,6,3) (1 case)
Total favorable outcomes =1+3+1=5
4. Final Probability
The probability P is the ratio of favorable outcomes to total outcomes:
P=Total OutcomesFavorable Outcomes
Final Answer:
The probability that the equation has equal roots is 2165.
