Calculate confidence interval for binomial distribution

I want to understand how to compute hypothesis tests for a binomial distribution. An example is something like

A diagnostic test should be correct in 96% of cases. For a sample of 75, 68 outcomes were correct. Does this result support a confidence level of 90% that the claim is true?

I can see that I can do that:

CDF[BinomialDistribution[75, 0.96], x]

and by trial and error you will find the values ​​of x that result in CDF values ​​of 5%, 95%. How do I get the correct value of x without trial and error?