[Math] If $a$ is an odd integer then $x^2+x-a = 0$ has no integer solutions

Question

I'm suppose to prove by contrapositive that if $a$ is an odd integer then the equation $x^2+x-a=0$ has no integer solution.

By contrapositive:

If the equation $x^2+x – a = 0$ has an integer solution then $a$ is an even integer. So I attempt to apply the quadratic formula and have this result $\frac{-1 \pm \sqrt{1 – 4a}} 2$. I have no idea how I'm suppose to get an integer solution from this, let alone an even solution. I've tried to multiply by the conjugate but it gets really messy and I feel that I'm over-thinking it. Thanks for your help.

Answer 1

Hint: If $x$ is any integer, then $x^2+x$ is even.