Let $G$ be a group of order $8$ and $x$ be an element of $G$ of order $4$. Prove that $x^2 \in Z(G)$

Question

I am quite a beginner in group theory, so I need to get my proofs checked. Here's what I've done:

We know, that the center of a group $G$, i.e $Z(G)$ is a normal subgroup of $G$ [no proof required here]. So, the quotient group $G/Z$ can be considered. Now, consider $x^2Z*x^2Z=x^4Z=Z=(x^2Z)^2$, which readily implies that $x^2 \in Z(G)$.

Is this all? Or am I missing something?