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

The Correct Option is B

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).