BURLINGAME, Calif. — This is the most populous state in the nation, but in presidential elections, it has almost always found itself on the sidelines. The last time a Republican primary in California mattered was 52 years ago, when the party nominated Barry Goldwater.

But California will finally play an important role in the drama of presidential politics.

The 172 Republican delegates up for grabs in the state’s June 7 primary, which in most years has taken place well after the party’s presidential standard-bearer was known, will determine whether Donald J. Trump can clinch the nomination on the last day of voting and avoid having to wrangle uncommitted delegates at the national convention in July.

All three Republican presidential candidates, along with hundreds of reporters, turned up for the party’s state convention here over the weekend, the biggest show in memory of presidential candidate firepower at this yearly gathering.

“California is at a crossroads — California is going to decide this Republican primary,” Senator Ted Cruz of Texas, who may need a win in Indiana on Tuesday to keep his hopes alive here, told hundreds of conventiongoers Saturday. “Who’d have thunk it?”