Let α and β be the roots of the quadratic equation x^2-2bx+c^2=0...

Given the quadratic equation $x^2 - 2bx + c^2 = 0$ with roots $\alpha$ and $\beta$.

Sum of the roots is $\alpha + \beta = 2b$.

Product of the roots is $\alpha \beta = c^2$.

The arithmetic mean $A$ of $\alpha$ and $\beta$ is given by $A = \dfrac{\alpha + \beta}{2} = \dfrac{2b}{2} = b$.

The geometric mean $G$ of $\alpha$ and $\beta$ is given by $G = \sqrt{\alpha \beta} = \sqrt{c^2} = c$ (since $c$ is a positive real number).

The second quadratic equation is $x^2 - (b+c)x + bc = 0$.

Factoring the equation, we get $(x - b)(x - c) = 0$.

The roots are $x = b$ and $x = c$.

Substituting $b = A$ and $c = G$, the roots are $A$ and $G$.

Answer: $A,\ G$