Given quadratic equation is

$$a x^2+b x+c=0 \text { where } a, b, c \in\{1,2,3, \ldots, 8\}$$

For repeated roots,

$$\begin{aligned} & b^2-4 a c=0 \\ & \Rightarrow b^2=4 a c \end{aligned}$$

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

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

Corresponding $$b$$ must lie in set $$\{1,2,3, \ldots 8\}$$

$$\begin{aligned} & (a, b, c) \in\{(1,2,1),(1,2,4),(2,4,2),(2,8,8) \\ & (3,6,3),(4,4,1),(4,8,4),(8,8,2)\} \\ & \therefore \text { probability }=\frac{8}{8^3} \\ & =\frac{1}{64} \\ & \end{aligned}$$