0

The problem states: If roots of $x^4 + ax^3 + bx^2 + ax + 1 = 0$ are real, find the minimum value of

$$a^2 + b^2$$

Im really not very good at these types of questions as they seemingly provide very little information.

So, I am basically stumped by this. I first thought that, since the roots are real, $x^4$ and $x^2$ should be positive, which would imply the others are negative. But wont know anything about $a$ or $b$, so really Ive no idea. Id like a hint and not the full solution, so that I can get better at solving such problems. Thanks.

Arthur
  • 199,419
MNIShaurya
  • 862

1 Answers1

1

Divide the equation by $x^2$ and set $y=x+\frac{1}{x}$.

flowsignal
  • 191
  • 1
    Only when $x\neq 0$. – amWhy Apr 03 '20 at 13:14
  • That's perhaps a first step or hint. So you get $y^2+ay+(b-2)=0$ must have two solutions with $y \in \mathbb R \setminus (-2, 2)$. From there you still need to reach the distant minimum for $a^2+b^2$... – Macavity Apr 03 '20 at 13:34