Algebra – Proving Product of Two Real Numbers is Maximum When Numbers Are Equal Given Constant Sum

Question

Let us consider two real numbers $x$ and $y$. How can we prove value of $xy$ is greatest when $x=y$ given the condition $x+y=$ constant?

I have already found a proof, but I am not entirely happy with it yet.

Answer 1

First prove that $$ xy = \frac{1}{4} \left( (x + y)^2 - (x - y)^2 \right)$$ How can you maximize $xy$ given this?