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:

$\begin{array}{cccc}& & (x-y{)}^2& \ge 0\\ ⟹& & x^2-2xy+y^2& \ge 0\\ ⟹& & x^2+2xy+y^2& \ge 4xy\\ ⟹& & \frac{x^2+2xy+y^2}{4}& \ge xy\\ ⟹& & {\left(\frac{x+y}{2}\right)}^2& \ge xy\\ ⟹& & \frac{x+y}{2}& \ge \sqrt{xy}\end{array}$

Note that the inequality at the start is an equality iff Template:Math, hence the arithmetic mean of two numbers is equal to the geometric mean of those numbers iff the two numbers are equal.