Question

If each of the coefficients \( a, b, c \) in the equation \( ax^2 + bx + c = 0 \) is determined by throwing a die, then the probability that the equation will have equal roots, is:

• A. \( \frac{1}{36} \)
• B. \( \frac{1}{72} \)
• C. \( \frac{7}{216} \)
• D. \( \frac{5}{216} \)

Hint:

To find the probability of equal roots in a quadratic equation, set the discriminant \( D = 0 \) and count integer solutions for \( b^2 = 4ac \).

Answer 1

The Correct Option is D

Step 1: Condition for Equal Roots
A quadratic equation \( ax^2 + bx + c = 0 \) has equal roots if and only if the discriminant is zero: \[ D = b^2 - 4ac = 0. \] Step 2: Total Possible Cases
Each coefficient \( a, b, c \) is determined by throwing a die, meaning each can take any value from \( 1 \) to \( 6 \).
Thus, the total number of possible choices for \( (a, b, c) \) is: \[ 6 \times 6 \times 6 = 216. \] Step 3: Counting Favorable Cases
We need to count the number of cases where \( b^2 = 4ac \). For each value of \( b \) (1 to 6), we check how many pairs \( (a, c) \) satisfy this equation:
- \( b = 1 \): No integer values of \( (a, c) \) satisfy \( 1 = 4ac \).
- \( b = 2 \): No integer values of \( (a, c) \).
- \( b = 3 \): \( 9 = 4ac \) leads to no integer solutions.
- \( b = 4 \): \( 16 = 4ac \) gives possible pairs \( (a, c) = (4,1), (2,2), (1,4) \) → 3 solutions.
- \( b = 5 \): No integer values of \( (a, c) \).
- \( b = 6 \): \( 36 = 4ac \) gives possible pairs \( (a, c) = (6,3), (3,6) \) → 2 solutions.
Total favorable cases: \[ 3 + 2 = 5. \] Step 4: Compute Probability \[ P(\text{equal roots}) = \frac{\text{favorable cases}}{\text{total cases}} = \frac{5}{216}. \] Thus, the correct answer is \( \mathbf{\frac{5}{216}} \).