If $3x^2+2ax+(a^2+2b^2+2c^2)=2(ab-bc)$, then a,b,c are in GP,HP or AP?

Question

I do not really see any condition placed on $a$, $b$ and $c$ from which we can deduce whether it is in GP,HP or AP. In the given question, nothing has been mentioned about $x$. So we can assume only that $x$ is a variable.When $x$ is a variable,that may take any value, how can I deduce a relation among $a$, $b$ and $c$? Where am I mistakened?Any help is very much appreciated.

Answer 1

We need $$a^2-3(a^2+2b^2+2c^2-2ab+2bc)\geq0$$ or $$\left(a-\frac{3b}{2}\right)^2+3\left(\frac{1}{2}b+c\right)^2\leq0,$$ which gives $a=\frac{3b}{2}$ and $c=-\frac{b}{2}$, which is nothing.

Maybe you mean $$3x^2+2ax+(a^2+2b^2+2c^2)=2(ab+bc)?$$ If so then $a$, $b$, $c$ is AP.