WEST PALM BEACH, Fla. — It’s either candidate’s race to win as President Barack Obama and Mitt Romney prepare for their second debate Tuesday night, with just three weeks to go until the election and voting well under way in many states.

The Republican challenger had trailed the Democratic incumbent in national polls for weeks, but now has drawn even, benefiting from a boost of enthusiasm following a strong first debate performance 10 days ago. While Romney’s standing has improved in some states, Obama retains an edge in the hunt for the 270 electoral votes needed to take the White House. The president also has far more ways than Romney to reach that magic number.

But that’s not enough to calm nervous Democrats, even as they revel in Vice President Joe Biden’s pull-no-punches turn on the debate stage Thursday night against GOP vice presidential nominee Paul Ryan. They are looking for an equally aggressive Obama to show up for the prime-time town-hall style debate in Hempstead, N.Y.

“The race is tightening,” said Mo Elleithee, a Democratic campaign strategist and former aide to Hillary Rodham Clinton during her primary campaign against Obama in 2008. “It will be very, very close.” But, he added, “The president will win re-election.”

Steve Schmidt, the chief Republican strategist four years ago for GOP nominee John McCain, acknowledged Obama’s edge but said it could be erased if the president comes off as defensive or dismissive in the second debate as he did in the first. “If he has another debate performance anywhere near that vicinity, it’s going to go south for him,” Schmidt said.

Last week’s feisty confrontation between Biden and Ryan set the stage for Tuesday’s presidential debate and gave Republicans an opening to intensify their criticism about Obama’s foreign policy. Romney has jumped on Biden’s assertion that “we weren’t told” of an official request for more security at a consulate in Libya that was attacked by terrorists who killed the U.S. ambassador and three other Americans. The White House spent the bulk of Friday trying to explain what Biden meant.

Expect that issue to come up Tuesday.

As the debate looms large as one of the final opportunities to affect the trajectory of the race, both campaigns are working feverishly in the nine most competitive states — Colorado, Florida, Iowa, Nevada, New Hampshire, North Carolina, Ohio, Virginia and Wisconsin — to get their core supporters to vote early and persuade undecided voters to back their candidate.

TV ads are a near constant presence, mailboxes are filled with campaign brochures and door-step visits by volunteers are picking up. Obama, Romney, their running mates, families and high-profile Democrats and Republicans are near constant presences in those states, working to tip the balance in a tight race where any factor could make a difference.

Romney’s biggest challenge remains Ohio, where polls show Obama with a consistent, slight lead.

Without Ohio’s 18 electoral votes, Romney would have to win nearly all the other contested states, most of which are either too close to project a winner or are leaning Obama’s way, some solidly.

Given the stakes, Romney spent the past week bearing down on Ohio, campaigning there four of five days last week and boosting his television advertising, according to ad-spending reports provided to The Associated Press. Both Romney and Ryan scheduled events in the state Saturday.

Ohio is proving to be tricky for Romney. The state has an unemployment rate lower than the national average and a revived energy sector built on natural gas. Also, Obama’s auto industry bailout is popular.

Plus, the president has kept his eye squarely on the state even as polls showed him in strong position. He has visited twice this month, and plans to return in the coming week. Obama also has kept pace with Romney’s Ohio ad spending.

Elsewhere, polls show the race a dead heat in Florida, the biggest prize up for grabs with 29 electoral votes, and Virginia, where Romney has posted gains over the past week. In North Carolina, polls also show the race close, although Obama has trimmed his advertising in recent weeks. Surveys show Obama leading in Colorado, Wisconsin and Iowa, and very narrowly in Nevada.

In the aftermath of Romney’s debate performance, Republicans have grown more enthusiastic, which is a critical development in the homestretch. A Pew Research Center poll last week found energy levels even for the first time, with 68 percent of registered voters who say they back Obama strongly supporting him and 67 percent of Romney voters strongly behind him.

A Washington Post-ABC News poll on voters favorability toward the candidates conducted after the first debate showed Romney viewed as “strongly favorable” by 62 percent of Republicans, his highest level in that poll.

Republicans are working to harness that enthusiasm to counter Obama’s tested early-voting program, which in 2008 made the difference when Obama carried Colorado, Florida, Iowa and North Carolina.

No votes will be counted until Nov. 6.

But early reports from North Carolina and Florida encourage GOP workers.

Among the 29,400 voters who have cast absentee ballots in North Carolina, 54 percent are registered Republicans and 28 percent are Democrats, according to the United States Elections Project at George Mason University. Democrats have a big lead in Iowa in terms of early ballots cast by party registration.

Obama’s voter registration push has resulted in more Democrats than Republicans on the voter rolls in most of the tightest states, including Florida and Nevada, according to data from state election boards. Even so, Republicans have narrowed the Democratic voter advantage since 2008 in many of the battlegrounds, including Iowa.