Solution:

Let a and b be the roots of the quadratic equation.

So, in accordance to the given condition, we have

$ A.M\text{ }=\text{ }\left( a+b \right)/2\text{ }=\text{ }A $

$ a\text{ }+\text{ }b\text{ }=\text{ }2A\text{ }\ldots ..\text{ }\left( 1 \right) $

$ GM\text{ }=\text{ }\surd ab\text{ }=\text{ }G $

$ ab\text{ }=\text{ }{{G}^{2}}\ldots \text{ }\left( 2 \right) $

The quadratic equation is given as follows:

x² – x (Sum of roots) + (Product of roots) = 0

x² – x (2A) + (G²) = 0

x² – 2Ax + G² = 0 [Using (1) and (2)]

Therefore, the required quadratic equation is x² – 2Ax + G² = 0.