Key Concepts and Formulas

• Discriminant of a Quadratic Equation: For a quadratic equation $Ax^2 + Bx + C = 0$, the discriminant is given by $\Delta = B^2 - 4AC$.
• Condition for Real Roots: A quadratic equation has real roots if and only if its discriminant is non-negative, i.e., $\Delta \geq 0$.
• Probability: For a finite sample space with equally likely outcomes, the probability of an event is given by $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$.

Step-by-Step Solution

Step 1: Understand the Sample Space The coefficients $a, b,$ and $c$ are the outcomes of three independent rolls of a fair tetrahedral die. A tetrahedral die has four faces marked $1, 2, 3, 4$. This means each of $a, b,$ and $c$ can take any integer value from the set $\{1, 2, 3, 4\}$. Since the rolls are independent, we can determine the total number of possible ordered triples $(a, b, c)$.

• Number of choices for $a$: 4 (1, 2, 3, 4)
• Number of choices for $b$: 4 (1, 2, 3, 4)
• Number of choices for $c$: 4 (1, 2, 3, 4)

The total number of possible outcomes in the sample space is the product of the number of choices for each variable: $\text{Total outcomes} = 4 \times 4 \times 4 = 4^3 = 64$

Step 2: Apply the Condition for Real Roots For the quadratic equation $ax^2 + bx + c = 0$ to have real roots, its discriminant must be non-negative. Here, $A=a$, $B=b$, and $C=c$. $b^2 - 4ac \geq 0$ We need to find the number of combinations $(a, b, c)$ from our sample space that satisfy this inequality. This will give us the number of favorable outcomes.

Step 3: Systematically Count Favorable Outcomes We will iterate through all possible values of $a$ (from 1 to 4), then $b$ (from 1 to 4), and for each pair, determine the valid values of $c$ (from 1 to 4) that satisfy $b^2 \geq 4ac$.

• Case 1: $a = 1$ The inequality becomes $b^2 \geq 4c$.

  • If $b=1$: $1^2 \geq 4c \Rightarrow 1 \geq 4c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=2$: $2^2 \geq 4c \Rightarrow 4 \geq 4c \Rightarrow c \leq 1$. So, $c=1$. (1 case: $(1,2,1)$)
  • If $b=3$: $3^2 \geq 4c \Rightarrow 9 \geq 4c \Rightarrow c \leq 2.25$. So, $c=1,2$. (2 cases: $(1,3,1), (1,3,2)$)
  • If $b=4$: $4^2 \geq 4c \Rightarrow 16 \geq 4c \Rightarrow c \leq 4$. So, $c=1,2,3,4$. (4 cases: $(1,4,1), (1,4,2), (1,4,3), (1,4,4)$) Total for $a=1$: $0 + 1 + 2 + 4 = 7$ favorable outcomes.

• Case 2: $a = 2$ The inequality becomes $b^2 \geq 8c$.

  • If $b=1$: $1^2 \geq 8c \Rightarrow 1 \geq 8c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=2$: $2^2 \geq 8c \Rightarrow 4 \geq 8c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=3$: $3^2 \geq 8c \Rightarrow 9 \geq 8c \Rightarrow c \leq 1.125$. So, $c=1$. (1 case: $(2,3,1)$)
  • If $b=4$: $4^2 \geq 8c \Rightarrow 16 \geq 8c \Rightarrow c \leq 2$. So, $c=1,2$. (2 cases: $(2,4,1), (2,4,2)$) Total for $a=2$: $0 + 0 + 1 + 2 = 3$ favorable outcomes.

• Case 3: $a = 3$ The inequality becomes $b^2 \geq 12c$.

  • If $b=1$: $1^2 \geq 12c \Rightarrow 1 \geq 12c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=2$: $2^2 \geq 12c \Rightarrow 4 \geq 12c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=3$: $3^2 \geq 12c \Rightarrow 9 \geq 12c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=4$: $4^2 \geq 12c \Rightarrow 16 \geq 12c \Rightarrow c \leq 4/3 \approx 1.33$. So, $c=1$. (1 case: $(3,4,1)$) Total for $a=3$: $0 + 0 + 0 + 1 = 1$ favorable outcome.

• Case 4: $a = 4$ The inequality becomes $b^2 \geq 16c$.

  • If $b=1$: $1^2 \geq 16c \Rightarrow 1 \geq 16c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=2$: $2^2 \geq 16c \Rightarrow 4 \geq 16c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=3$: $3^2 \geq 16c \Rightarrow 9 \geq 16c$. No $c \in \{1,2,3,4\}$ satisfies this. (0 cases)
  • If $b=4$: $4^2 \geq 16c \Rightarrow 16 \geq 16c \Rightarrow c \leq 1$. So, $c=1$. (1 case: $(4,4,1)$) Total for $a=4$: $0 + 0 + 0 + 1 = 1$ favorable outcome.

Step 4: Calculate the Total Number of Favorable Outcomes Summing the favorable outcomes from all cases: $\text{Total favorable outcomes} = 7 + 3 + 1 + 1 = 12$

Step 5: Calculate the Probability The probability $P$ is the ratio of favorable outcomes to the total possible outcomes: $P = \frac{12}{64}$ To simplify the fraction, divide both numerator and denominator by their greatest common divisor, which is 4: $P = \frac{12 \div 4}{64 \div 4} = \frac{3}{16}$

Step 6: Determine $m$, $n$, and their Sum The probability is given as $\frac{m}{n}$ where $\operatorname{gcd}(m,n)=1$. From our calculation, $m=3$ and $n=16$. We check that $\operatorname{gcd}(3,16)=1$, as 3 is prime and 16 is not a multiple of 3. Finally, we need to find $m+n$: $m+n = 3 + 16 = 19$

Common Mistakes & Tips

• Systematic Counting: The most common error is missing cases or double-counting. Always use a structured approach (e.g., iterating through $a$, then $b$, then $c$) to ensure accuracy.
• Inequality Interpretation: Be careful when converting inequalities like $c \leq 2.25$ into possible integer values within the given range $\{1,2,3,4\}$.
• Arithmetic Errors: Double-check calculations, especially squaring and division, as small mistakes can propagate and lead to an incorrect final count.

Summary This problem required applying the discriminant condition for real roots of a quadratic equation ($b^2 - 4ac \ge 0$) to a probability scenario. We first determined the total sample space of $4^3 = 64$ possible $(a,b,c)$ combinations from the tetrahedral die rolls. Then, through a systematic case-by-case analysis, we counted the 12 combinations that satisfied the real roots condition. The probability was found to be $\frac{12}{64}$, which simplifies to $\frac{3}{16}$. Thus, $m=3$ and $n=16$, leading to $m+n=19$.

Final Answer The final answer is $\boxed{19}$.