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 :
For the quadratic equation \( ax^2 + bx + c = 0 \), where \(a, b, c\) are selected from the set \(\{1, 2, 3, 4, 5, 6\}\).
Step 1: Real Roots Condition Real roots exist if the discriminant, \( D = b^2 - 4ac \), is non-negative (\( D \geq 0 \)).
Step 2: Total Combinations The total number of possible combinations for \((a, b, c)\) is \( 6 \times 6 \times 6 = 216 \).
Step 3: Probability and Count Let \(N\) be the number of combinations satisfying the conditions for real roots where one root is larger than the other. The probability \(p\) is \( p = \frac{N}{216} \). Consequently, \( 216p = N \). It is given that \( N = 38 \).
