So I have noticed that when you press [ALPHA][ENTER] (labeled SOLVE) it seems to have the exact same function as pressing enter. It will enter an expression on the home screen in the same way as just plain [ENTER] and gores to a new line in the program editor just the same as well. Is there any special function that this has? It has been bugging me that I can't figure it out of there is one. Here is a picture of it:

It's used in the [MATH][0] solver and (I believe) nowhere else.

I actually never realized [ALPHA][ENTER] was intended to be used for the solver. Thanks a lot!

I don't know how the solver works, but I can see that [SOLVE] does something special. This is kind of changing the subject, but can somebody explain how to use the solver/what it does? I am confused as to how to use it/what to use it for.

Basically, the solver (at the bottom of the [MATH] menu) allows you to enter an equation set equal to zero, and then solve for x. you use [solve] after you type a guess into the X= bar, which will then display the closest solution your guess that solves the equation.
On the 83's and 84 plus's the solve function only allows you to set it equal to zero, but the newer CE ( and probably the CSE, but I am not sure) allows you to set it equal to something besides zero, i.e. x^2+4x+3=x+4.
The solve function is a very powerful that often goes unused function of these calculators. I have always thought that teachers should especially be informed about this feature because they try to take all of the necessary precautions in clearing your calcs before algebra tests, but many of them don't eve realize that they all have built in quadratic solvers and more.
Of well. That is all you need to know about the solve function.

Yeah essentially, when in the solve menu, you use the ALPHA/ENTER to get a solution to your equation. I think this is because you can use the regular ENTER to move to the other arguments of the solve( command (the bounds, the guess, etc.)
Also, CHill, the root that it spits out is not necessarily the closest one to your guess although it usually is. It's just that the calculator uses Brent's method to solve equations numerically, which requires an initial guess, like newton's method, but that guess can actually make it converge on a root that is not the one closest to your guess (assuming there are multiple roots). This is most likely to occur in the bad guess zones which are usually near the local mins and maxes, where the first interpolation will throw you far away from the initial guess.