Can anyone prove the following equation, occurring while optimizing an n-body program?

floor(sqrt(2k) + 1/2) =?= floor(sqrt(2k-3/4)+1/2) for all k integer >= 1

floor is the integer part of the argument. It is false for some k real >=1

Extensive numerical checks seems to confirm it, except for some very large k the numerical check was finding the equation false, but this could be ascribed to round-off errors.

I would be grateful knowing what are the general methods that experts in number theory would use to prove this kind of equation.