Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$
What's the minimum value of $(a^2+b^2)$?
Ok so here I tried to divide by $x^2$ and get this : $$x^2+\frac{1}{x^2}+a\left(x+\frac{1}{x}\right)+b=0$$ Here with a subtle $t$ substitution: $$x+\frac{1}{x}=t\Rightarrow x^2 + \frac{1}{x^2}=t^2-2$$ $$t^2-2+at+b=0$$
And right where I thought, that I had tackled the problem, here I get stuck. I don't know if this is the right approach or not but since it required the minimum value I tried to turn the equation into some sort of inequality where later on, terms would get cancelled and I could end in a $(a^2+b^2)\ge$ and determine the minimum value.
Any help is appreciated.
