Is there a way to solve for when some function returns an integer?

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I currently have a function (single variable) that is under a square root, and I want to find when the value of the function returns an integer value, in terms of the variable.

Like if I took f(x) = SQRT(x+1) , I want it to return as like x = C^2 + 2C. I see that its like completing the square except the function under the square root, but is slightly too complicated to do by hand. I was trying to use Mathematica, but I don't think I'm doing something wrong.