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:

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

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.