Is there a way to prove that the results of quadratic equation are always odd?

Question

Given the following quadratic equation:

$$f(x) = x^2-3x-1$$

For $x, 1\leq{x}\leq{100}, x \equiv 1\mod 2$

Is this true for all $x \in{\mathbb{Z}}$? If so, is there a way to prove it?

Answer 1

If $x = 2k + 1$, then

$$x^2 - 3k - 1 = (2k + 1)^2 - 3(2k + 1) - 1 = 2k(2k - 1) -3 \equiv 1 \pmod{2},$$

so the result is always odd.