Question

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 ______.

Answer 1

Correct Answer: 19

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, \( m + n = 19 \)

Answer 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.