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.