Proving an Inequality using a Different Method Is there another way to prove that: If $a,b\geq 0$ and $x,y>0$
$$\frac{a^2}{x} + \frac{b^2}{y} \ge \frac{(a+b)^2}{x+y}$$
using a different method than clearing denominators and reducing to $(ay-bx)^2 \ge 0$? 

One method of solving any inequality is to determine points where the associated equality is true or where the functions involved are not continuous.  That is because, as long as $a$ and $b$ are continuous, we can only go from "$a> b$" to "$a< b$", or vice-versa, by going through "$a= b$".
However here, the simplest way to solve the equality is by "clearing denominators" so apparently you would not consider that a "different" way.
