Is $p\neq 0$, then there are $x_+, x_- \in \mathbb{C}$ with $x_\pm ^2=p$.
My proof:
The proof for $p>0$ is trivial, since $\mathbb{R}\subset \mathbb{C}$ and it's true $\forall p\in \mathbb{R}$.
The proof for $p<0$: Since $i:=\sqrt{-1}$ and $p$ would be a multiple of $\sqrt{-1}$ since $\sqrt{p}$ with $p>0$ can be rewritten als $\sqrt{-1\cdot (-p)}=\sqrt{-1}\cdot \sqrt{-p}$