Prove that all solutions to the equation x² = x +1 are irrational

Question

I'm pretty new to discrete mathematics and I'm having a hard time understanding how to solve some problems--this one in particular.

Any help would be greatly appreciated even if it's just something to help me get started. Thanks.

Answer 1

By the Rational Root Theorem, the only possible rational solutions are $-1, +1$, which are promptly seem not to be solutions.

Answer 2

Do you know that for a rational root $p/q$ of a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ with integer coefficients

• $p$ must divide $a_0$, and
• $q$ must divide $a_n$ ?

Answer 3

Any solution of $x^2=x+1$ is a solution of $x=1+\frac{1}{x}$, hence a number with an infinite continued fraction representation, hence an irrational number.