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

It is given that the A.M of the roots $=8$ and their G.M.$=5$

We know that the A.M between $a$ and $b=\large\frac{a+b}{2}$ and

G.M.$=\sqrt {ab}$

$\Rightarrow\:\large\frac{a+b}{2}$$=8$ and $\sqrt {ab}=5$

$\Rightarrow\:a+b=16$ and $ab=25$

Since $a$ and $b$ are roots of the quadratic equation,

sum of the roots $=a+b=16$

Product of the roots $=ab=25$

We know that the general quadratic equation is

$x^2-$(Sum of the roots)$x+$(product of the roots)=0.

$i.e.,$ The required quadratic equation is $ x^2-16x+25=0$