Arithmetic and geometric mean/Estimate/Exercise/Solution

The square of any real number is non-negative, performing some algebraic manipulations allows us to derive the arithmetic mean--geometric mean inequality as follows:

${\begin{aligned}&&(x-y)^{2}&\geq 0\\\implies &&x^{2}-2xy+y^{2}&\geq 0\\\implies &&x^{2}+2xy+y^{2}&\geq 4xy\\\implies &&{\frac {x^{2}+2xy+y^{2}}{4}}&\geq xy\\\implies &&\left({\frac {x+y}{2}}\right)^{2}&\geq xy\\\implies &&{\frac {x+y}{2}}&\geq {\sqrt {xy}}\end{aligned}}$

Note that the inequality at the start is an equality iff x = y, hence the arithmetic mean of two numbers is equal to the geometric mean of those numbers iff the two numbers are equal.