Heading west from the hologram fountain where you speak with Elijah, refer to your Pip-boy local map and make your way to the Medical District. Once you enter, Elijah will warn you about the speakers found in the area. There is one ahead of you, above an archway, which can be safely shot from a distance.
There are several ghost people in the area. You can choose to fight them or carefully sneak past.
Go through the archways linking each open area. When you approach the archway with a toppled trash can in front of it, your collar will start to beep. Run straight to the closed door visible ahead. If you turn to face the way you came you will see the offending speaker high above the archway you ran through, and you can now safely destroy it.
The closed door hides a cloud-filled room. You can explore it now if you have enough health items to avoid death. Dean Domino'sUnclean Living companion perk may prove useful here.
Continue following your map marker to the Villa clinic main entrance. You can also enter via the upper floor, by making your way through the nearby buildings, but note that this leads you past more speakers and puts you close to a hologram guard in the clinic. Using the main entrance will give you a chance to get your bearings and collect some useful items before proceeding.
Entering the clinic (through either door) will prompt a further message from Elijah, warning about the holograms. There is a terminal near each entrance that contains further information and allows you to reprogram the hologram, which is patrolling the upper floor when you enter.
To deactivate the hologram you can disable (Repair 25) or destroy its emitter unit. This is located on the upper floor, on the wall halfway along the corridor. It glows bright blue and is easily visible. Use the terminal to set the patrol route to the lower floor, or simply stay out of view Hidden behind the hologram and follow it along the corridor until you reach the emitter. With the hologram gone, the only threat remaining in the clinic is the shielded speaker downstairs.
From the main entrance, there is a restroom behind the desk with a water source (irradiated). The left doorway leads upstairs.
Through the right doorway, a room to your right contains headless corpses, and a terminal with information that adds to the History's Sake challenge. The corridor straight ahead leads to the basement (to your left) and the other stairway leading upstairs (to your right). Down the corridor to your left is the shielded speaker which Elijah will advise you to disable.
The speaker terminal is down in the clinic basement past a locked door (Lockpick 75). You can find the key in one of the upstairs rooms in a desk (along with the Med-X vending machine code).
Once in the basement, access the terminal to shut down the speaker, allowing you to safely reach Christine.
Head back upstairs and return to the now deactivated speaker. Turn left and enter the end room, which contains a malfunctioning auto-doc. Activating this machine will release Christine, and after a short conversation in which you explain the situation, she will agree to join you. You can have her accompany you (if you have no other companion with you), or send her to wait at the fountain. Once this conversation ends the quest will be complete.
Once this quest is completed, several ghost people will appear in the Medical District outside the clinic. Christine does not have any weapons when released from the auto-doc, so if she is your companion you may want to arm her before leaving the clinic.