Integral Solutions of $x+y=x^2−xy+y^2$

Question

Find all integral solutions of $x+y=x^2−xy+y^2$

In this answer, in the last step(of 1st answer) if $-3x^2+6x+1\ge 0$ how did they find the values of $x$ and how to prove that they are the only ones ?

Answer 1

It's just a matter of finding the roots of the quadratic. $$-3x^2+6x+1\ge0\implies3x^2-6x-1\le0$$ The leading term is positive and the inequality points left, so we know that the range of satisfying $x$ is finite. The roots are $1\pm\frac2{\sqrt3}$, and $x=0,1,2$ are the only integers between those roots, as said in the linked answer.

Answer 2

Because the leading coefficient of the given quadratic equation is negative, the equation attains negative values for $x$ greater than both roots or less than and positive values for $x$ between the roots. In this particular case, there happens to be three integers between the roots, therefore three integers where the quadratic is non negative.