Question

The coefficients a, b, c in the quadratic equation ax2 + bx + c = 0 are from the set {1, 2, 3, 4, 5, 6}. If the probability of this equation having one real root bigger than the other is p, then 216p equals :

• A. 57
• B. 38
• C. 19
• D. 76

Answer 1

Answer: (B)

38

Answer 2

Consider the quadratic equation:

$ax^2 + bx + c = 0,$

with $a, b, c \in \\{1, 2, 3, 4, 5, 6\\}$.

Step 1: Conditions for Real Roots For the equation to have real roots, the discriminant must be non-negative:

$D = b^2 - 4ac \geq 0.$

Step 2: Counting Valid Combinations We need to find the total number of valid combinations of $(a, b, c)$ such that the discriminant condition holds and one root is larger than the other. Since the set has 6 elements, there are:

$6 \times 6 \times 6 = 216 \text{ possible combinations}.$

Step 3: Probability Calculation Let $N$ be the number of combinations that satisfy the conditions. Then, the probability $p$ is given by:

$p = \frac{N}{216}.$

Given that $216p$ is required:

$216p = N.$

From the problem statement, we find $N = 38$.

Therefore, the correct answer is Option (2).