When will the arithmetic and geometric means be equal

note that x(n) is x subscript n
so x(1) is x sub 1 which is just first number
and x(2) is x sub 2 which is just 2nd number

arithmetic mean

x(1) + x(2)
▬▬▬▬▬
.......2

geometric mean
.._______
\/x(1)x(2)

set these equal

x(1) + x(2)........._______
▬▬▬▬▬..=..\/x(1)x(2)
.......2


multiply both sides by 2
......................________
x(1)+x(2) = 2.\/x(1)x(2)

Remove radical by calculating the square of both sides


[x(1) + x(2)]²..=..4x(1)x(2)

which is same as

[x(1)]²+2x(1)x(2) + x(2)]²..=..4x(1)x(2)

subtract 4x(1)x(2) from both sides to set quadratic
equal to zero

[x(1)]²..-..2x(1)x(2) + [x(2)]²..=..0


Since there are two variables it is tough to
solve for any roots so I will add 2x(1)x(2) to
both sides to observe

[x(1)]² + [x(2)]²..=..2x(1)x(2)

These are equal whenever

x(1) = x(2)

and both are non negative.

that is x(1) ≥0 and x(2)≥0