What you have so far looks good. You already know that $b + d = 1$, so the goal would be to find what $a + c$ is.

Note that $a$ is an integer and, since $x > 2$, you have that $a \geq 1$. Since $|1 - a|+\sqrt{c - 4} = 1$, what can you say about $|1 - a|$? Use these to find the possible values of $a$. Then use those to find possible values of $c$ in each case.