In order to use the Luck option, Speech skill of 65 is also required. Otherwise, Caesar is not convinced and he will not allow you to operate on him. Surgical tools will not appear in New Vegas Medical Clinic and you will be forced to retrieve the part in Vault 34.
With either of the above methods, you have the option to kill Caesar during the procedure. To remain on good terms with Caesar's Legion after his death, you must pass a skill check (Speech 50, Medicine 50, or a high Legion reputation) when speaking to Lucius.
If Arcade Gannon is an active follower, you have the option to sell him into slavery as Caesar's personal doctor (Karma loss), thereby completing the quest. To do so, have him wait outside Caesar's tent. Talk to Lucius inside.
If you choose to find the diagnostic scanning module, you must go to the radiation and ghoul infested Vault 34. You can find the vault north-northwest of the NCR's Camp Golf or east of the NCR's Sharecropper Farms near New Vegas. It is recommended that you bring a good supply of RadAway and Rad-X. There will be golden geckos on the way in as well as the ghouls inside.
Simply go to the room marked 'Clinic'.
As you walk in take your first right to the stairway that will take you to the 'cafeteria'.
Drop down the hole, exit the room and turn left.
Follow the corridor (and signs for the 'clinic'), until you see the room marked 'Clinic'.
You need to go into one of the submerged rooms and search one of the dead vault technicians for the 'Vault 34 Security Terminal Password'.
You will need to find a security terminal and go to the option 'System' and then 'Activate Pump Station'.
Go through the Clinic and follow the corridor until you reach a door that was previously closed on the right side of the hallway leading to the 'Armory'.
Once in the door make the first right and enter the 'Security Station'.
Once in there, take a right, and try to sneak if you can. (You may need more Rad-X and RadAway), as there is a lot more radiation down here.
Attempting the surgery without the required Medicine 75 or Luck 9 will result in Caesar's death. You must then have either Medicine 50 or Speech 50 to explain to Lucius that Caesar's age was a complication in the procedure, or he will become hostile (unless your reputation with the Legion is "Liked" or higher).
In all versions of the game, the broken Auto-doc is in the clinic in Vault 34, not Security Station A.
You will not be able to offer Arcade as a slave if you recruited him after you have spoken to Caesar in private.
Behind the scenes
The title of this quest is a play on Julius Caesar's famous last words,"Et tu, Brute?", from Shakespeare's play Julius Caesar which translates to "And you, Brutus?", showing Caesar's dismay in seeing Brutus taking part in his assassination.
The quest is an elaborate allusion to Julius Caesar, who did indeed suffer from an ailment that caused severe headaches and was also a secret from the public. Caesar's headaches were more likely to have been caused by a motor function disorder, causing him to fall regularly ("falling sickness").
If after Arcade is given as a slave the player speaks to him and says "The world needs Caesar more than it needs you" Arcade will respond "Victrix causa diis placuit sed victa Catoni" which can be translated as "the victorious cause pleased the gods, but the conquered cause pleased Cato." This is a quote from the Latin poem Pharsalia by Lucan, in which the historical figure of Caesar is portrayed as the main antagonist, and the main hero, the historical figure of Cato Uticensis, is eventually forced to commit suicide, being unable to defeat Caesar.