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rationals wanted, a\alpha^2 + b\alpha+ c = \sqrt{d} if \alpha^3 = \alpha + 1

Source: 2015 VMEO IV Juniors 10.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

September 15, 2021
algebrarational

Problem Statement

Given a real number α\alpha satisfying α3=α+1\alpha^3 = \alpha + 1. Determine all 44-tuples of rational numbers (a,b,c,d)(a, b, c, d) satisfying: aα2+bα+c=d.a\alpha^2 + b\alpha+ c = \sqrt{d}.
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