Here, the difference between equation 1 and equation 2 makes a difference I never would have predicted:

In[1]:= $Assumptions = x \[Element] Reals && p \[Element] Reals;

In[2]:= equation1 := E^{I answer x} == E^{I p x};

In[3]:= equation2 := E^{I answer x} == E^{-I p x};

In[4]:= Solve[Eliminate[equation1, x], answer]

Out[4]= {{}}

In[5]:= Solve[Eliminate[equation2, x], answer]

During evaluation of In[5]:= Eliminate::ifun: Inverse functions are being used by Eliminate, so some solutions may not be found; use Reduce for complete solution information.

Out[5]= {{answer -> -p}}

In[6]:= Reduce[Eliminate[equation1, x], answer]

Out[6]= True

In[7]:= Reduce[Eliminate[equation2, x], answer]

During evaluation of In[7]:= Eliminate::ifun: Inverse functions are being used by Eliminate, so some solutions may not be found; use Reduce for complete solution information.

Out[7]= answer == -p || p == 0 || x == 0

In[1]:= E^{I answer x^2} == E^{-I x^2/(4 p)};

In[2]:= Eliminate[%, x]

During evaluation of In[2]:= Eliminate::ifun: Inverse functions are being used by Eliminate, so some solutions may not be found; use Reduce for complete solution information.

Out[2]= 4 answer p Log[E^((I x^2)/(4 p))] == -((I x^2)/(4 p))

In[3]:= PowerExpand[%]

Out[3]= I answer x^2 == -((I x^2)/(4 p))

In[4]:= SolveAlways[%, x]

Out[4]= {{answer -> -(1/(4 p))}}

The "Eliminate" doesn't actually eliminate x, but the answer depends on that step, for some reason that eludes me.

Thanks.

Thank you. I'm getting good use out of this program, and this is just my first week using it.

Thank you. That is a good pattern-matching solution I had overlooked. However, it succeeds by easing the burden on Solve, but I'm trying to get a better handle on using Solve on more difficult problems (i.e., involving complex exponentials / branch-cuts). I'm still curious how I could properly call Solve to do it, without my passing Solve a simplified problem, as your solution does. I'm sure Solve (or Reduce) can do the whole thing, as the answer I received from Solve appears essentially correct (but it contains more information than I'm seeking).