<s>[INST] Prove that $a^2+ab+b^2\ge 0$ How to prove that $a^2+ab+b^2\ge 0$?
Obviously the squares are positive, but how can I be sure that $ab$ doesn't become too negative with a certain combination of $a$ and $b$?
 [/INST] $a^2+ab+b^2= (a+ \frac{1}{2}b)^2 + \frac{3}{4}b^2$
One may prefer symmetry:
$2(a^2+ab+b^2)= a^2 +b^2+(a+b)^2$.
</s>