Let \(a\), \(b\), and \(c\) denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that \(ax^2 + bx + c = 0\) has all real roots is \(\frac{m}{n}\), \(\text{gcd}(m, n) = 1\), then \(m + n\) is equal to ______.
Correct Answer: 19
Approach Solution - 1
Given that \( a, b, c \in \{1, 2, 3, 4\} \)
Consider the quadratic equation:
\( ax^2 + bx + c = 0 \)
For the equation to have all real roots, the discriminant must be non-negative:
\( D \geq 0 \Rightarrow b^2 - 4ac \geq 0 \)
Case 1: Let \( b = 1 \)
\( 1 - 4ac \geq 0 \Rightarrow \) Not feasible.
Case 2: Let \( b = 2 \)
\( 4 - 4ac \geq 0 \Rightarrow 1 \geq ac \)
Hence, \( a = 1, c = 1 \)
Case 3: Let \( b = 3 \)
\( 9 - 4ac \geq 0 \Rightarrow \frac{9}{4} \geq ac \)
Possible pairs: \( (a, c) = (1, 1), (1, 2), (2, 1) \)
Case 4: Let \( b = 4 \)
\( 16 - 4ac \geq 0 \Rightarrow 4 \geq ac \)
Possible pairs: \( (a, c) = (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2) \)
Total valid combinations: \( 12 \)
Total possible outcomes: \( (4)(4)(4) = 64 \)
Therefore, the probability is:
\( P = \frac{12}{64} = \frac{3}{16} = \frac{m}{n} \)
Hence, \( m + n = 19 \)
Approach Solution -2
Step 1: Conditions for real roots For the quadratic equation \( ax^2 + bx + c = 0 \) to have all real roots, the discriminant \( D \) must satisfy:
\[ D \geq 0. \]
The discriminant is given by:
\[ D = b^2 - 4ac. \]
Step 2: Values of \(a, b, c\) Since \( a, b, c \) are outcomes of three independent rolls of a tetrahedral die, their possible values are:
\[ a, b, c \in \{1, 2, 3, 4\}. \]
Step 3: Solve for \( b^2 - 4ac \geq 0 \) We analyze cases for \( b \):
- Case 1: \( b = 1 \)
- Case 2: \( b = 2 \)
- Case 3: \( b = 3 \)
- Case 4: \( b = 4 \)
Step 4: Total favorable outcomes The total number of favorable outcomes is:
\[ 1 + 3 + 8 = 12. \]
The total possible outcomes are:
\[ 4 \times 4 \times 4 = 64. \]
Step 5: Probability The probability is:
\[ P = \frac{12}{64} = \frac{3}{16}. \]
Step 6: Simplify \( m + n \) Here:
\[ m = 3, \quad n = 16, \quad m + n = 19. \]
Final Answer: 19.
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