Inequality with Pythagorean Triples

Question

When finding the Pythagorean triple for $a + b + c = 1000$, with $a < b < c$ and of course $a^2 + b^2 = c^2$, Wolfram alpha gives me $$ a < -500(\sqrt2 - 2) $$ My question is where does this come from? Thanks!

Answer 1

Substituting $c=1000-a-b$ into $a^2+b^2=c^2$ yields $$(1000-a)(1000-b)=500000.$$ That entails $1000-a>\sqrt{500000}$ , that is: $$a<1000(1-1/\sqrt2),$$ which is exactly the given inequality.