An inequality to find the range of an unknown coefficient Find the range of $a$ such that 
$$a(x_1^2+x_2^2+x_3^2)+2x_1x_2+2x_2x_3+2x_1x_3 \geq 0, x_i\in \mathbb{R}$$
I tried to use Cauchy Inequality but it seems not...

Answer: $a \ge 1$
Let $a=1:$ $$x_1^2+x_2^2+x_3^2+2x_1x_2+2x_2x_3+2x_1x_3 =(x_1+x_2+x_3)^2\ge 0$$
If $a>1$ then $$(a-1)(x_1^2+x_2^2+x_3^2)+x_1^2+x_2^2+x_3^2+2x_1x_2+2x_2x_3+2x_1x_3 \ge 0$$
If $a<1, a=1- E$ then 
$(x_1+x_2+x_3)^2-E(x_1^2+x_2^2+x_3^2)<0$ at $x_1=-x_2-x_3$
