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

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.