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After subjecting the nation’s biggest banks to the most public scrutiny in decades, federal regulators ordered 10 of them on Thursday to raise a total of $75 billion in extra capital and gave the rest a clean bill of health.

The long-awaited results of the “stress tests,” a central part of the Obama administration’s plan to restore the financial industry to health, set off an immediate scramble by major institutions for more capital, which they must raise by November.

The verdict was far more upbeat than many in the industry had feared when the tests were first announced in February. And the banks that came up short will have to raise much less than some analysts had expected as recently as a few days ago.

The stress tests were aimed at estimating how much each bank would lose if the economic downturn proved even deeper than currently expected. But regulators gave the banks a break by letting them bolster their capital with unusually strong first-quarter profits and also by letting the banks predict modest profits even if the economy again turns down.

Despite an almost tangible sense of relief among the banks and investors, the report card is unlikely to silence an intense debate over whether the Treasury Department and the Federal Reserve let the banks off too easily and glossed over their real problems.

Under the worst-case situation — an unemployment rate of 10.3 percent next year, an economic contraction of 3.3 percent this year and a 22 percent further decline in housing prices — the losses by the 19 banks could total $600 billion this year and next, or 9.1 percent of the banks’ total loans, regulators concluded. That rate of loss would be higher than any other since 1921, including the losses during the Great Depression.

But while the adverse situation was supposed to be unlikely, it is not that much worse than what has happened so far. Unemployment hit 8.5 percent in April and could top 9 percent as early as Friday, when the Labor Department releases its employment report for May.

Bank of America was told it would have to come up with $34.9 billion. Wells Fargo will have to find $13.7 billion. And Citigroup must produce another $5.5 billion, on top of the $45 billion that it had already planned to acquire by letting the Treasury become its biggest single shareholder.

Industry executives reacted with jubilation, as if they had proved their critics wrong and passed the tests with flying colors.

“The results off the stress tests should put to rest the harmful speculation we have seen over the past few months,” declared Edward L. Yingling, president of the American Bankers Association, hours before the results were even made public.

Investors, who had already bid up share prices of the big banks in reaction to leaks about the results earlier this week, reacted with relief.

Regulators and bank executives alike predicted that most of the institutions would be able to build up the necessary capital from private sources — either by selling off assets or by converting shares of preferred stock into ordinary stock.