Question

The coefficients a, b, c in the quadratic equation ax² + bx + c = 0 are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is :

• A. \(\frac{3}{256}\)
• B. \(\frac{1}{128}\)
• C. \(\frac{1}{64}\)
• D. \(\frac{3}{128}\)

Answer 1

The Correct Option is C

Given the quadratic equation: \[ ax^2 + bx + c = 0 \] where \(a, b, c \in \{1, 2, 3, 4, 5, 6, 7, 8\}\).

For repeated roots, the discriminant must be zero: \[ D = 0 \implies b^2 - 4ac = 0 \implies b^2 = 4ac \]

The total number of possible choices for \((a, b, c)\) is: \[ 8 \times 8 \times 8 = 512 \]

Number of favorable cases for \(b^2 = 4ac\) is 8. Therefore, the probability is: \[ \text{Prob} = \frac{8}{512} = \frac{1}{64} \]

The possible values for \((a, b, c)\) satisfying \(b^2 = 4ac\) are: \[ (1, 2, 1), \, (2, 4, 2), \, (1, 4, 4), \, (4, 4, 1), \, (3, 6, 3), \, (2, 8, 8), \, (8, 8, 2), \, (4, 8, 4) \] This gives 8 cases.