Irrational and rational proofs

Hey, I was wondering about proving that square roots of numbers are rational or irrational.
I know about proving this for the numbers 1(duh), 2 and all multiples of 2 and 4, but what about 3?
How do i prove that 3(or any other number) is irrational or rational?

assume x = sqrt(3) is rational
x^2 = 3
x^2 - 3 = 0

use the rational roots theorem to find the only possible rational solutions for this equation are 1,-1,3,-3, none of which solve the equation.  so, no rationals solve this equation.

thus sqrt(3) is not rational since it solves the equation but there is no rational solution.