Values of $x_1^2 + x_1 x_2 + x_2^2$

Question

What values can the expression $x_1^2 + x_1 x_2 + x_2^2$ take, depending on parameter $a$, where $x_1$ and $x_2$ ($x_1\not=x_2$) are two roots of the following equation. $$x^3-2007x=a$$

It's obvious that if $a=0$ then $x_1=0$ or $x_2=0$ $\Rightarrow$ expression can be equal $0$. But if $a\not=0$ I don't really know what to do. Well, after differentiation and dividing by 3 I got $$x^2-669=0$$ hence when $x\in[-\sqrt{669}; +\sqrt{669}]$ function plot is decreasing $\Rightarrow$ I should find out extreme conditions according to the point of tangency of that "hill". And I don't know how to do that

Answer 1

If $x_1,x_2$ are distinct roots of the polynomial $x^3-2007x-a$, then we have $$ x_1^3 = 2007x_1+a,\qquad x_2^3=2007 x_2 +a $$ from which $$ x_1^2-x_2^3 = 2007 (x_1-x_2) $$ and by dividing both sides by $x_1-x_2$ (which we assumed to be $\neq 0$) we get $$ x_1^2+x_1 x_2+x_2^2 = \color{red}{2007}.$$

Answer 2

Since $x_1+x_2+x_3=0$, it's just $$\frac{1}{2}(x_1^2+x_2^2+(x_1+x_2)^2)=$$ $$=\frac{1}{2}\sum_{cyc}x_1^2=\frac{1}{2}((x_1+x_2+x_3)^2-2(x_1x_2+x_1x_3+x_2x_3))=2007$$

Answer 3

Hint: $x_1^3-2007 x_1 = x_2^3-2007 x_2 \implies x_1^3-x_2^3=(x_1-x_2)(x_1^2+x_1 x_2 + x^2)=2007(x_1-x_2)$.