If α is a root of $x^2-x-1=0$, then α³ expressed in the form pα + q is:

The equation says α² = α + 1. Multiplying both sides by α gives α³ = α² + α, and replacing α² once more gives α³ = (α + 1) + α = 2α + 1. (α + 2 stops after one substitution too few; 3α + 1 adds an extra α; 2α − 1 slips a sign.) With α = (1 + √5)/2 ≈ 1.6180, α³ ≈ 4.2361 and 2α + 1 ≈ 4.2361.