If the arithmetic mean of two numbers is 15 and their geometric mean is 9 find the two numbers

Answer 1

(a+b)/2 = 15
√(ab) = 9
so,
ab = 81
The only factors of 81 are (1,81),(3,27),(9,9)

Clearly the only possible candidates are 3 and 27, which you can easily verify.

If you want to do the algebra, you have

a+b = 30
a^2+2ab+b^2 = 900

a^2 + 2*81 + (81/a)^2 = 900
a^2 + 6561/a^2 = 738
a^4-738a^2+6561 = 0
(a^2-9)(a^2-729) = 0
a = 3 or 27