About the Golden Ratio Conjugate

Question

My question is quite simple: I have the Golden Ratio, noted "φ", and the so-called Silver Ratio, noted "ψ". Both of these numbers are solutions for the equation x² - x - 1 = 0, and I want to demonstrate that ψ=1-φ, and ψ=-1/φ. I just know that (x-φ)(x-ψ)=x² - x - 1, but that's all, I just don't get it although it seems very obvious. Any advice on this? Thanks for your replies!

Answer 1

HINT: Taking up from where you have left,

$$(x-\phi)(x-\psi)=x² - x - 1$$ $$\implies x^2-(\phi+\psi)x+\phi\psi=x² - x - 1$$

Now, just compare the coefficients of the terms on either side of the above equation and re-arrange it to get what you want to demonstrate.

For further reference, you should check Vieta's Theorem.

Answer 2

Ah, yes, I think I understand! So I have −(ϕ+ψ)=−1, and it makes -ψ=-1+ϕ, which is equivalent to ψ=ϕ-1, right? Same for ψ=-1/ϕ, I just have to look at ϕψ=-1. I didn't know this theorem, I will remember it for later! Thank you a lot :)