To find the probability that the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are equal when \( a, b, c \) are the outputs obtained from rolling three unbiased dice, we need to follow these steps:
### Step 1: Understand the Condition for Equal Roots
The roots of the quadratic equation are equal if the discriminant is zero. The discriminant \( D \) is given by:
\[
D = b^2 - 4ac
\]
For the roots to be equal, we need:
\[
b^2 - 4ac = 0 \implies b^2 = 4ac
\]
### Step 2: Possible Values of \( a, b, c \)
Since \( a, b, c \) are the outcomes of rolling three unbiased dice, each can take any integer value from 1 to 6. Therefore, the possible values for \( a \) and \( c \) must be considered to satisfy the equation \( b^2 = 4ac \).
### Step 3: Analyze the Equation \( b^2 = 4ac \)
We can rewrite the equation as:
\[
ac = \frac{b^2}{4}
\]
This means \( b^2 \) must be divisible by 4, which implies \( b \) must be even. The possible even values for \( b \) from the outcomes of rolling a die are 2, 4, and 6.
### Step 4: Case Analysis for Each Even Value of \( b \)
#### Case 1: \( b = 2 \)
\[
b^2 = 4 \implies 4ac = 4 \implies ac = 1
\]
The only possibility is \( (a, c) = (1, 1) \). This gives us 1 combination.
#### Case 2: \( b = 4 \)
\[
b^2 = 16 \implies 4ac = 16 \implies ac = 4
\]
The pairs \( (a, c) \) that satisfy \( ac = 4 \) are:
- \( (1, 4) \)
- \( (4, 1) \)
- \( (2, 2) \)
This gives us 3 combinations.
#### Case 3: \( b = 6 \)
\[
b^2 = 36 \implies 4ac = 36 \implies ac = 9
\]
The only pair that satisfies \( ac = 9 \) is:
- \( (3, 3) \)
This gives us 1 combination.
### Step 5: Total Combinations
Now, we add the number of combinations from all cases:
- From Case 1: 1 way
- From Case 2: 3 ways
- From Case 3: 1 way
Total combinations = \( 1 + 3 + 1 = 5 \).
### Step 6: Total Sample Space
When rolling three dice, the total number of outcomes is:
\[
6 \times 6 \times 6 = 216
\]
### Step 7: Calculate the Probability
The probability \( P \) that the roots of the equation are equal is given by:
\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{216}
\]
### Final Answer
Thus, the probability that the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are equal is:
\[
\frac{5}{216}
\]
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