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By Susan Page, USA Today–

Donald Trump and Hillary Clinton did what they had to do to put their campaigns back on track Tuesday: win big on their home turf.

Their muscular victories in the New York primary didn’t eliminate concerns about Trump’s toxicity in the Republican Party or about Clinton’s vulnerabilities in the Democratic race. And their top challengers, Texas Sen. Ted Cruz and Vermont Sen. Bernie Sanders, vowed to continue campaigns that have become more difficult but are still caustic.

Still, the New York primary steadied the campaigns of the leading candidates after both hit a speed bump two weeks ago with losses in the Wisconsin primary. And their leads could be fortified if they manage to replicate the victories in five Northeast primaries next week.

In their victory speeches, both Trump and Clinton signaled they were eager to turn to the general-election campaign.

Surveys of voters as they left polling places made it clear why, underscoring some of the scars the primary battles are leaving. One in four Sanders supporters said they wouldn’t vote for Clinton in the general election if she won the Democratic nomination. Four in 10 Republican voters said they would be “scared” or “concerned” if Trump were elected president.

Nearly six in 10 Republicans said the battle had divided the party, not energized it.

That said, Trump’s triumph was so resounding that the TV networks declared him the winner as soon as the polls closed at 9 p.m.

“We’re going to be so great again, and I just can’t wait,” the billionaire businessman declared in a victory speech at Trump Tower in Manhattan, where he had announced his campaign almost a year ago. He predicted Cruz’s challenge was now all but over. “We don’t have much of a race anymore.”

“There’s no place like home,” Clinton said in her victory speech. She said the Democratic nomination fight was “in the homestretch, and victory is in sight.”

Actually, Trump still could fall short of the 1,237 he needs to prevail on the first ballot at the Republican convention this summer. Clinton is on track to win a delegate majority at the Democratic convention, but probably not until the final big primary day on June 7. For the first time in four decades, neither party has a presumptive nominee at this point in the season — that is, a candidate who commands a majority of convention delegates or has seen the last serious challenger withdraw.

Next week’s contests could be part of a crucial two-step, with primaries in five states that share a geographic neighborhood and political terrain with New York. More delegates are at stake in those combined contests — in Connecticut, Delaware, Maryland, Pennsylvania and Rhode Island — than in New York.

In Pennsylvania and Maryland, the states with the biggest prizes, Trump and Clinton have double-digit leads, according to recent statewide polls averaged by realclearpolitics.com. More limited polling in Connecticut and Rhode Island give Trump an overwhelming edge and Clinton a narrower one.

Braced for defeat in New York, the challengers on Tuesday already were looking past it. Sanders campaigned in Pennsylvania, where he drew thousands on the campus at Penn State. Ohio Gov. John Kasich stumped in Pittsburgh, then headed to Maryland. Cruz spent Tuesday night in Philadelphia.

All insisted they were in the race for the long haul. None of them were pulling their punches.

In a radio interview with Sean Hannity, Cruz accused the Trump campaign of “whining” about the delegate selection process — the Texas senator has exploited the rules in a string of states — and added, “I cannot help that the Donald Trump campaign does not seem capable of running a lemonade stand.”

At a town hall in Annapolis, Kasich accused Trump of making promises he couldn’t keep. “OK, everybody here is going to be a billionaire,” he joked.

At his Penn State rally, Sanders mocked Clinton, saying that her remarks before Wall Street firms “must be a speech written in Shakespearean prose” to be worth the six-figure checks she was paid for them.

For native-son Trump and former senator Clinton, New York really did turn out to be home sweet home, quieting questions about their campaigns.

Neither Trump nor Clinton had scored a victory since the Arizona primary on March 22. Cruz had won five in a row since then; Sanders had carried seven of the previous eight state contests.

But if Clinton can dominate next week’s contests on top of her victory in New York, she would need win fewer than one in four of the remaining delegates. If Trump can sweep next week’s contests on top of his victory Tuesday, he would need to win about half of the remaining delegates — a higher bar, but not an impossible one.

What’s more, there’s this: More than seven in 10 Republican voters in New York said the candidate leading in delegates should get the nomination, even if he’s short of a majority.

That’s an argument Trump has been making for weeks.