The coefficients a , b , c in the quadratic equation a x 2 + b x + c = 0 are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is: [2024]

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Solution

Q.
The coefficients $a, b, c$ in the quadratic equation $a x^{2} + b x + c = 0$ are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is: [2024]

1 $\frac{3}{256}$

2 $\frac{1}{128}$

3 $\frac{1}{64}$

4 $\frac{3}{128}$

Ans.
(3)

For equation $a x^{2} + b x + c = 0$ to have repeated roots we have

$D = b^{2} - 4 a c = 0$

$\Rightarrow b^{2} = 4 a c = 2^{2} (a c)$

$\Rightarrow a c$ must be a perfect square.

$\Rightarrow (a, c) = {(1, 1), (1, 4), (2, 2), (2, 8), (3, 3), (4, 1), (4, 4), (5, 5), (6, 6), (7, 7), (8, 2), (8, 8)}$

$∴ (a, b, c) = {(1, 2, 1), (1, 4, 4), (2, 4, 2), (2, 8, 8), (3, 6, 3), (4, 4, 1), (4, 8, 4), (8, 8, 2)}$

Required probability $= \frac{8}{8^{3}} = \frac{1}{64}$

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