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

Answer: (C)

$\frac{1}{64}$

Answer 2

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.