Question

For y ∈ Z, prove that 9y² + 3y − 2 is even.

Answer 1

Consider two cases:

1. Suppose that $y$ is odd. It means, that we can present $y$ as: $y=2n+1$, where $n\in\mathbb{Z}$. We get: $9(2n+1)^2+3(2n+1)-2=9(4n^2+4n+1)+6n+1=36n^2+42n+10$. The latter number is even
2. Suppose that $y$ is even. It means that we can present $y$ as: $y=2n$. We receive: $36n^2+6n-2.$ The latter number is even.

Thus, we have shown that the number $9y^2+3y-2$ is even.