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

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.