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…
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 :
$\frac{1}{128}$
$\frac{1}{64}$
$\frac{3}{256}$
$\frac{3}{128}$
Solution
$\begin{aligned} & a x^2+b x+c=0 \\ & a, b, c \in\{1,2,3,4,5,6,7,8\} \\ & \text { Repeated roots } D=0 \\ & \Rightarrow b^2-4 a c=0 \Rightarrow b^2=4 a c \\ & \text { Prob }=\frac{8}{8 \times 8 \times 8}=\frac{1}{64} \\ & \Rightarrow(a, b, c) \\ & (1,2,1) ;(2,4,2) ;(1,4,4) ;(4,4,1) ;(3,6,3) ; \\ & (2,8,8) ;(8,8,2) ;(4,8,4) \\ & 8 \text { case }\end{aligned}$