Eureka!

Eureka! is the final quest in Fallout: New Vegas if you choose to side with the NCR.

Description
Take over the Hoover Dam with the NCR.

Quick walkthrough

 * 1) Talk to General Oliver.
 * 2) (Optional) Flush the Legion invaders through the turbines.
 * 3) Make your way through the power plant and across the dam.
 * 4) Arrive at the Legion camp across the bridge, speak to Legate Lanius.
 * 5) By passing a Speech or Barter check (100), the player can convince the Legate not to fight.
 * 6) Otherwise, the player will have to defeat the Legate.
 * 7) Once defeated, speak with General Oliver.

Detailed walkthrough
After completing You'll Know It When It Happens, at the very moment you choose to side with the NCR for the final battle (thus completing For the Republic, Part 2), you'll be automatically taken from Col. Moore's office to General Oliver's Compound, where the General will begin briefing you. The player is given the chance to wait, as this begins the end quest for the game. If the player chooses to continue, after the briefing, Legion members attack the Hoover Dam and you have to fight your way through the power plant rooms and head towards their base east. You will start in Power Plant 04. Completing this will require manual intervention; return to Power Plant 04 by the closest door which is right below the console. On the east wall, on your left as you re-enter the area, there is an exit to the Dam exterior. The manual release valve you need to complete this part of the quest is along the river to your left between the broken retainer wall, just after the second sandbag wall. Return to Power Plant 03.
 * (Optional) Mike Lawson will speak to you. If you wish to follow his suggestion and do the optional part which involves flushing the Legion invaders through the turbines, agree to his suggestion. To complete the optional part of the quest, follow the staircase immediately to the left leading back upstairs to enter Power Plant 03. The console is on the second floor of Power Plant 03, on the southern wall. Passing a Science check (65) allows the player to use the terminal to engage the turbines.

Proceed to Power Plant 02, then Power Plant 01, dispatching Legion troops on your way. Climb the stairs on your left in Power Plant 01, then proceed north along the balcony. Follow the passage along the left, then go up the stairs, and exit via the elevator to the Hoover Dam visitor's center. At this point, you will begin to notice assistance appearing from companions and factions you have helped out along the way. Exit via the double doors on the east side to Hoover Dam.

Proceed to the north along the curved walkway above the dam. There is a door hidden in the side of the dam that you can pick with a lockpick skill of 75 (hard). Enter the Hoover Dam checkpoint. From here, you can check the terminal; if you have a science skill of 100 and a detonator, you may use the terminal to detonate one group of legions (the blast is centered near the first tower). When you exit on the other side. An NCR Veteran Ranger will thank you for your support, asking how to coordinate their assistance. The player can respond with several dialogue choices:
 * Rangers can snipe Legion Soldiers for you;
 * Rangers will follow you;
 * With a sufficient Speech skill (85), the player can persuade The NCR Veteran Ranger to make the Rangers pave the way for you;
 * Rangers will fall back, providing no assistance.

After crossing the dam, head left along the bridge to the east, and continue, following the road. Enter the gate to the Legate's Camp. Head east into the camp, following the path up the side of the ridge. There, you will encounter Legate Lanius. Note that With a Speech or Barter skill of 100, you can pass a series of skill checks, and avoid fighting him. The Legate is extremely challenging, and many Legionaries will come out and attack you. If you pass all of the previous speech checks and fail the last one, you get another Speech challenge [85] which can convince the Legate to fight you one on one - all the other Legion troops will become friendly to you.
 * There is another tree of answers in dialogue with Lanius - the speech checks that suggest bluffing. This way is actually easier, because it does not require level 100 Speech (checks are 60-75-90-90). In the last Speech check all three answers give the same result, but after it you must choose the answer, and the Legate will flee.

After either killing the Legate, or passing a series of speech checks to end the fight peacefully, you should notice the gate to the camp has closed. Head back to it. The gate will explode and you will meet with General Oliver.

Behind the scenes
"Eureka" is the state motto of California. The word comes from the ancient Greek word, heúrēka, meaning "I found it". There is a story of the famous Greek mathematician Archimedes, who shouted eureka upon making a great discovery and ran naked through the streets of Syracuse.

Bugs

 * Occasionally, leaving Power Plant 1 without completing the optional quest marker, you will randomly fail 'Don't Tread on the Bear!' and all NCR forces will go hostile to you.
 * If you stay back until the Legate has killed both NCR Rangers, he will not talk to the player on his own. That gives you the possibility for a sneak kill
 * Sometimes the Legate will not register as dead and you cannot continue with the quest. POSSIBLE Fix: If you go through the speech checks and convince him to fight you one-on-one, he will register as dead and you can continue the quest. Killing all the other Legion troops might fix this problem as well.
 * Using spawned none of the needed minions to continue the mission regardless of the fact that the player has progressed and done all available side missions and fought through to the legates camp (as the previous mission had bugged).
 * If the final missions have bugged out while trying to help the NCR (as happens if you complete Wild Card: Side Bets before For The Republic Part 2), you can use . You then fight your way to the legate camp but Legate Lanius won't appear. You can then use and General Lee will appear as you go to leave the legate camp, thereby starting the endgame sequence. I have tried spawning a Legate Lanius but he is immediately hostile and set to essential so he's not much use.
 * Doors may require key and cannot leave Visitor Center. SOLUTION: apparently if the staging area is left too soon after the president's departure, the doors will not unlock. They appear to unlock shortly before the "Quest Completed" finishes fading from the screen. a safe bet would be to save just after the president finishes the speech, and checking the doors before continuing onto another area.
 * The Legate has been known to flee once the two NCR Rangers appear, he will go out of the main gate, and you cannot follow him as the gate will have no sense of interaction.
 * There is a glitch where you will be able to fast travel out of the dam and camp, but cant reenter being the area is sealed off. Making it impossible to finish the quest. SOLUTION: a weak solution to this problem: If you want to fast travel, you must enter an area that you can not fast travel from (house, building, tent, etc.) and exit it again. Immediately upon loading outside, you must press the pip-boy button and you will be able to fast travel, but then you must repeat the processes again after arriving at the next destination. This allows you to continue the quest.
 * It is possible for a companion to become effectively trapped in Oliver's compound. this is due to the fact it become labeled "locked requires key". This means the trapped companion cannot follow you. for some odd reason they will not teleport as usual when you go through a loading screen door. This was experienced with Veronica Santangelo and Rex as a companion.Rex followed as normal.
 * Even after you kill Legate the quest marker will remain on him until all of Caesar's Legion in the area are dead. This can some times break the quest as the legion in the area are effectively stored in an unreachable area to the left of the main building were you encounter legate. This can be fixed via explosives such as rockets to kill them via blast radius.
 * It is possible for the player to fast travel when this quest is in progress by simply going through the hoover dam checkpoint and straight after you come out quickly go to your pipboy and fast travel. WARNING after doing this you will have to use that same method if you want to fast travel again.
 * It is possible that, while fighting the Legate, a glitch may cause him to suddenly "fly" straight up into the air almost beyond the point of visibility before plummeting back down to the ground, killing him on impact, thus winning the fight and the game for you. (This glitch was only observed after many, many failed attempts at killing him and restarting from the last save-point.)
 * If you are using weapons with high rates of fires, being around others with high ROF weapons may cause severe frame rate lag or freezes.

Эврика!