The question is as follows:
Let $C \in\mathbb {Z^+.}$ Prove that the complex roots of the equation $X^2 + X + \sqrt{C}=0$ are quadratic $\iff $ $C$ is a square in $\mathbb {Z}$.
I have attempted this question for quite some time without getting anywhere, my notes do not seem to be assisting me. Any help would be greatly appreciated!
Quadratic integers are roots of an equation $x^2+ax +b=0$ where $a$ and $b$ are integers.
If $C$ is an integer square the condition is automatically satisfied.
Conversely suppose that $X^2 + X + \sqrt{C}=0$ and $X^2+AX +b=0$. Subtracting we have $(A-1)X=\sqrt C-b $ and then $X$ would be real unless $A=1$ and $\sqrt C-b=0. $
Therefore $C=b^2$ as required.