MathLabs

Problem 3

Determine the minimum of a2+b2a^2+b^2 over real a,ba,b for which x4+ax3+bx2+ax+1=0x^4+ax^3+bx^2+ax+1=0 has at least one real solution.
Step 1 of 5: Reduce the quartic
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In plain words

The reciprocal symmetry of the quartic suggests the substitution.

y=x+1/xy=x+1/x
Detailed analysis

Since x≠0x\ne0, division by x2x^2 gives y2+ay+b−2=0y^2+ay+b-2=0. For real xx, necessarily ∣y∣≥2|y|\ge2.