Problem 3

Determine the minimum of

a^2+b^2

over real

a,b

for which

x^4+ax^3+bx^2+ax+1=0

has at least one real solution.

Step 3 of 5: Parameterize by the real root

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In plain words

Instead of imposing a possibly misleading linear constraint on $(a,b)$ , use the actual value of $y$ and express $b$ in terms of $a$ and $y$ .

b=2-y^2-ay,\quad a^2+b^2=a^2+(2-y^2-ay)^2

Detailed analysis

The equation $y^2+ay+b-2=0$ is equivalent to $b=2-y^2-ay$ . Thus, for any feasible pair $(a,b)$ and its corresponding real $y$ with $|y|\ge2$ , the objective is the displayed expression.