If AM and GM of two positive numbers a and b are 10 and 8 respectively,
Question:

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

Solution:

$\mathrm{AM}=10$

$\therefore \frac{a+b}{2}=10$

$\Rightarrow a+b=20 \quad \ldots \ldots(\mathrm{i})$

Also, $G=8$

$\therefore \sqrt{a b}=8$

$\Rightarrow a b=8^{2}$

$\Rightarrow a b=64$     ….(ii)

Using (i) and (ii):

$\Rightarrow a(20-a)=64$

$\Rightarrow a^{2}-20 a+64=0$

$\Rightarrow a^{2}-16 a-4 a+64=0$

$\Rightarrow a(a-16)-4(a-16)=0$

$\Rightarrow(a-16)(a-4)=0$

$\Rightarrow a=4,16$

If $a=4$, then $b=16$.

And, if $a=16$, then $b=4$.