Eureka!

Eureka! is the final quest in Fallout: New Vegas if the Courier sides with the NCR. Completing it unlocks the achievement of the same name.

Detailed walkthrough
After completing You'll Know It When It Happens, at the very moment the Courier might choose to side with the NCR for the final battle (thus completing For the Republic, Part 2), the player character will be automatically taken from Col. Moore's office to General Oliver's compound, where the general will begin a briefing. The Courier is given the chance to wait, as this begins the end quest for the game. If the player character chooses to continue, after the briefing, Legion members attack the Hoover Dam and one will need to fight their way through the power plant rooms and head east toward their base, starting in power plant 04.
 * (Optional) Mike Lawson will speak to the Courier. If they 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 in power plant 04 then across the room 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 character to use the terminal to engage the turbines. Completing this will require manual intervention; return to power plant 04 by the closest door which is right below the console. On the second floor on the east wall, from where the Courier reentered there is an exit to the Dam exterior. The manual release valve one needs to complete this part of the quest is along the river to the left between the broken retainer wall, just after the second sandbag wall. Return to power plant 03.

Proceed to power plant 02, then power plant 01, killing any Legion troops on the way. Climb the stairs on the left in power plant 01, then proceed north across the balcony. Follow the passage along the left, then go up the stairs, and exit via the elevator to the Hoover Dam visitor center. At this point, one will begin to notice assistance appearing from companions and factions the Courier has 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. Enter the Hoover Dam checkpoint. From here, check the terminal; if the player character has a Science skill of 100 and a detonator, they may use the terminal to detonate one group of Legion soldiers (the blast is centered near the first tower). When one exits on the other side, they will encounter a squad of NCR Veteran Rangers. The commander will thank the Courier for their support and ask how to coordinate their assistance.

The player character can respond with several dialogue choices which lead to the following actions:
 * The Rangers snipe Legion soldiers;
 * The Rangers follow the Courier into battle;
 * The Rangers will pave the way for the player character (requires a Speech check of 85);
 * 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, one will encounter Legate Lanius. Note that with a Speech or Barter skill of 100, one can pass a series of skill checks, and avoid fighting him. The Legate is extremely challenging, and many Legionaries will come out and attack to aid him. If one passes all of the previous Speech checks and fails the last one, the player character will get another Speech challenge (85) which can convince the Legate to fight them one on one - all the other Legion troops will become non-hostile.
 * There is another tree of answers in dialogue with Lanius - the speech checks that suggest bluffing. This way 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, one must choose the lie answer, and the Legate will flee.

After either killing the Legate or passing the Speech checks to end the fight peacefully, the Courier should notice the gate to the camp has closed. Head back to it. The gate will explode and the player character will meet with General Oliver.

Behind the scenes
"Eureka" is the state motto of California. The word comes from the ancient Greek word heúrēka, translated to "I have found."