For example, would solving for $x$ in $x^2=8x+7$ be the same as finding the roots of the equation? Also, would finding the roots of this be the same as finding the zeros?
For the following pattern:
currently, the preferred usage is (in order):
"zeros of an equation" and "solutions to a function" are just plain wrong.
Presumably if one is talking about an equation, then 'roots' is preferred (or rather, it is just how people happen to speak).
This is all to say that these phrases are all interchangeable, but some are preferred to others in their contexts.