Problem one: Even in his own party, a significant number of voters don’t believe Biden has accomplished very much.

Second, across the board, voters worry about the health and stamina of the 80-year-old president, the oldest to hold the office.

A third problem has no achievable solution: In today’s highly partisan times, nearly half of the country won’t favor the incumbent almost regardless of what he does. Over the last six presidential elections, only one — President Obama’s reelection in 2012 — featured a popular vote margin greater than 5%; another close contest in 2024 seems all but inevitable.

Winning in that environment doesn’t allow much margin for error. A president needs to hold the support of the vast majority of his own party and win over the relatively small but crucial share of voters who swing back and forth between the two sides.

One big speech can’t negate all of his challenges, but it can open the way, especially among members of the president’s Democratic Party. And those who watched Tuesday night — as well as the larger audience that will see snippets in the days to come — saw an event that went off without a major hitch.

Biden, who has struggled with stuttering since childhood, stumbled occasionally, as he always has. But he commanded the podium for well over an hour, parrying occasional heckling from Republicans such as Georgia’s Rep. Marjorie Taylor Greene and offering a list of nonpartisan proposals — fighting cancer, cracking down on so-called junk fees, helping veterans — that at times elicited nods of support from House Speaker Kevin McCarthy (R-Bakersfield), sitting behind him.

With at least a dozen calls for bipartisanship, which bracketed two direct veto threats — one against any abortion ban, the other against efforts to repeal last year’s Inflation Reduction Act — Biden sought to present himself as the mature adult keeping watch over an unruly Congress.

Those moments, especially the threat to veto an abortion ban, drew strongly positive responses from a panel of about 30 independent and nonpartisan voters convened in Las Vegas by Navigator Research, a Democratic firm.

The group showed “clear movement in favor of the president,” said Democratic pollster Margie Omero. Compared with the ratings that members of the group gave before the speech, “views of the president himself, his favorability, jumped about 20 points,” she said.

It’s important not to overstate how much difference that can make — memories of any single speech fade quickly.

And the challenges Biden faces remain significant. Within his own party, voters have been tepid, at best, about the idea of a second term. In a recent poll for the Associated Press conducted by NORC at the University of Chicago, just 37% of Democrats said they wanted him to run again. Among independents, only 12% were in favor of Biden running.

But the reaction to his speech Tuesday does illustrate that despite the problems he faces, Biden continues to have the ability to unify his party and reach a significant slice of independent voters.

That’s one major reason Biden has a strong shot at winning a second term. He also benefits from the risk Republicans face — a divisive primary with former President Trump threatening not to support the party if someone else becomes its nominee.

The biggest potential upside for Biden, however, is the possibility of significant economic improvement over the 21 months between now and the next election. Inflation already has been on the decline for months, and last week’s jobs report showed unemployment dropping to the lowest rate in more than 50 years.

“I would argue the Biden economic plan is working,” the president said after those numbers were released.

So far, voters’ view of the economy remains much more negative than the official statistics, but Democrats hope to turn around that skepticism.

“We’ve been though some really tough things” over the last few years, Omero said, so “it’s understandable that people feel wary of any signs of recovery.”

If current trends continue, some of that skepticism will fade, Democrats hope.

To take full advantage of any good economic news, however, Biden needs to persuade voters that he has put policies in place that mattered. So far, he hasn’t convinced them.

A recent Washington Post/ABC News poll illustrated the depth of that problem. Asked how much Biden had accomplished, 62% of respondents said he had achieved little or nothing, compared with 36% who said he had achieved a great deal or a good amount.

Republicans overwhelmingly fell into the negative camp, but even among Democrats, roughly 1 in 5 said he had achieved little or nothing. Among independents who lean Democratic, 30% took that view.

Biden and his aides vehemently dispute that idea, reeling off a list of achievements at every opportunity — the bipartisan infrastructure bill that will pump $1.2 trillion into repairing roads, bridges and rail systems, upgrading water systems to eliminate lead pipes and make other improvements; efforts to combat climate change; legislation to lower the price of prescription drugs for Americans covered by Medicare; and healthcare measures that have made insurance cheaper for millions of middle-income Americans.

Administration officials argue, with considerable justification, that their victories have been drowned out by the din of partisan warfare in Washington and by media coverage that focuses on conflict more than results.

It’s also true, however, that much of that new legislation has yet to have direct impact on Americans’ lives. The infrastructure law, for example, will play out over the next decade. The plan to reduce drug costs for seniors will start this year, with a $35 cap on the cost of insulin, but price reductions on a much wider range of drugs won’t start phasing in until 2026.

Biden conceded that point in a line in his speech that was not in his prepared text, saying that “so many things that we did are just now coming to fruition.”

Emphasizing his accomplishments has been a major preoccupation for Biden ever since the midterm campaign ended, with events like a trip to Baltimore late last month to tout funding to rebuild a critical railroad tunnel that dates to the Ulysses S. Grant administration, and another earlier in the month in Kentucky, where he and Senate Republican leader Mitch McConnell heralded plans to build a new bridge across the Ohio River.

That same list of victories formed most of the first half of Tuesday’s speech, underscoring how crucial it is for Biden to convince Americans that he has achieved victories.

Doing so is important to boost enthusiasm among Democrats, who want to see movement toward progressive goals, and to win over independents, who dislike partisan bickering.

It’s also critical to convincing Americans of one of the central claims of Biden’s presidency, said veteran Democratic strategist Steve Schale, a former Biden campaign advisor: “That there is a way to govern in this almost ungovernable moment.”