<s>[INST] Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution I just wanted the solution, a hint or a start to the following question.
Determine the minimum of $a^2 + b^2$ if $a$ and $b$ are real numbers for which
the equation
$$x^4 + ax^3 + bx^2 + ax + 1 = 0$$
has at least one real solution.
Thanks in advance.
 [/INST] As has already been observed, if $c$ is a root of your equation, so is $c^{-1}$. So suppose your equation has roots $c,d,c^{-1},d^{-1}$.
We have
$$a=-(c+c^{-1} + d + d^{-1})$$
and
$$b=cc^{-1}+ cd+cd^{-1} +c^{-1}d +c^{-1}d^{-1} + dd^{-1}$$
$$=2+(c+c^{-1})(d+d^{-1})$$
Now if $c\in \mathbb{R}$, we must also have $d+d^{-1}\in \mathbb{R}$. This means $d=i$ or $d\in \mathbb{R}$.
Thus if we write $z=c+c^{-1}$ and $w=d + d^{-1}$, we see that your problem is a question of minimising
$$f=(w+z)^2 + (2+wz)^2$$
subject to the constraint $|w|\geq 2$ and $|z|\geq 2$ or $z=0$.
If $z=0$ the $f=w^2+4\geq8$. Consider instead $|z|\geq 2$. In this case $|zw|\geq 4$ so $f\geq (2-zw)^2\geq 4$. Thus we see that the minimal value of $f$ is 4. This is obtained when $z=2,w=-2$. And the equation is
$$x^4-2x^2+1$$
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