Discrete Math "if and only if" Proof with Polynomials

Question

I'm doing my proof homework and we recently learned how to do "if and only if" proofs, but most of them are dealing with an even or an odd integer. This one states "Let n be an integer. Then 2n^2 - 3n - 2 = 0 if and only if 3n^2 - 7n + 2 = 0." Through my work I found that 2 is an acceptable integer, but we need to arbitrarily choose an integer. Is there a special circumstance for polynomials? Thanks.

Answer 1

Note that the only integer which satisfies $2n^2-3n-2=0$ is $n=2$. The other solution is $-1/2$ which is not an integer.

Also the only integer which satisfies $3n^2-7n+2=0$ is $n=2$. The other solution is $1/3$ which is not an integer.

Therefore for an integer $n$ the first statement is true if and only if the second is true.