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The tricky task now for all the GOP candidates is to brandish their conservative bona fides in the nomination fight in a way that they don’t turn off the broader electorate.

“Even if every Republican votes for you in a general [election], it is still not enough to get a majority of the votes,” Mr. Goldford said.

Former Utah Gov. Jon Huntsman Jr. echoed that sentiment in a recent interview with a warning: “When you find yourself in an extreme end of the Republican Party, you make yourself unelectable.”

“We’ve got to remember in order to beat Obama in 2012, we’ve got to have numbers on our side,” Mr. Huntsman said, casting himself as the more electable alternative to Mrs. Bachmann and Mr. Perry. “We’ve got to win back the Reagan Democrats. We’ve got to win back the disaffected independents who used to be Republicans.”

Going after independents and some votes in the opposing party usually is a general election strategy. Mrs. Bachmann, however, has made the claim early in the primary campaign as she tries to show GOP voters that her appeal is not limited.

Analysts said she is likely overstating the amount of support she has gained.

“It’s doubtful she has any meaningful support among Democrats or independents,” political analyst Charlie Cook said. “So, if she says she has the support of Republicans, Democrats and independents, it must be technically true. She has to have some support among at least a few, even if it implies something more than that.”

He added, “It’s kind of like her promise to get oil prices under $2 if elected. There has never been a four-year period when gas hasn’t gotten below $2, so the promise might not be as wild as it sounds.”