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Step-by-Step Solution
Step 1: Understand the total number of possible outcomes
Since the coefficients a, b, and c are determined by rolling a fair die, each can take any value from {1, 2, 3, 4, 5, 6}. Therefore, the total number of possible (a, b, c) combinations is:
$6 \times 6 \times 6 = 216$.
Step 2: Condition for the roots to be equal
For the quadratic equation $ax^2 + bx + c = 0$ to have equal roots, its discriminant must be zero:
$$
D = b^2 - 4ac = 0 \quad \Rightarrow \quad b^2 = 4ac.
$$
Step 3: Find all combinations satisfying $b^2 = 4ac$
We check which values of b (from 1 to 6) make $b^2$ divisible by 4, and then see for which (a, c) we have $ac = \dfrac{b^2}{4}$:
b = 2: $b^2 = 4$, hence $ac = \dfrac{4}{4} = 1$.
• Possible pairs of (a, c) are: (1, 1).
(Total 1 possibility)
b = 4: $b^2 = 16$, hence $ac = \dfrac{16}{4} = 4$.
• Possible pairs of (a, c) are: (1, 4), (4, 1), (2, 2).
(Total 3 possibilities)
b = 6: $b^2 = 36$, hence $ac = \dfrac{36}{4} = 9$.
• Possible pair of (a, c) is: (3, 3).
(Total 1 possibility)
Summing these up gives us:
$$
\text{Favorable cases} = 1 + 3 + 1 = 5.
$$
Step 4: Calculate the probability
The probability is the ratio of favorable cases to the total number of possible outcomes:
$$
\text{Required Probability} = \frac{\text{Favorable cases}}{\text{Total cases}} = \frac{5}{216}.
$$
Thus, the probability that the quadratic equation has equal roots is $ \frac{5}{216} $.
