Truth be told, we had given up any hopes of the original Moto X at Verizon and AT&T gaining root access on Android 5.1 update. But thanks to awesome developer community Android enjoys, it is real: You can now root your Moto X on Android 5.1 update, both on AT&T and Verizon.
That said, let us tell you that this Android 5.1 root is pretty dangerous, meaning it’s got more chances of bricking your devices and regular root tricks we come across daily.
During the course of root, your AT&T/Verizon Moto X will get kind-of soft bricked, getting stuck in HS-USB QDLoader 9008 mode. You will have to recover from it by installing the HS-USB QDLoader 9008 driver manually. And then run the root script to successfully root your Moto X.
→ This is only for Verizon Moto X (model no. XT1060) and AT&T Moto X (model no. XT1058). Other Moto X sets could be Unlocked using Motorola’s unlock service, and then rooted by flashing SuperSU using TWRP recovery, which can be installed on unlocked devices easily.
As regards how to root Verizon and AT&T Moto X, check out this thread.