Prove that there's a unique integer n where : $ 2n^2 − 3n − 2 = 0$.

Question

I have to prove that there's a unique integer n where : $2n^2 − 3n − 2 = 0$.

I factorised and got $(2n+1)(n-2) = 0$

I can then find both $n$ solutions : $-1/2$ and $2$. We can therefore see there's only 1 solution that is an integer.

The problem I have is that I think I'm missing something important here and I need confirmation.

Answer 1

You have solved the quadratic in the reals, which are a proper superset of the integers. You have shown there are only two real solutions, which immediately establishes there are at most two integer solutions. Furthermore, by inspection, you've concluded that one solution is not an integer, while the other is. This is sufficient to conclude there is exactly one integer solution.