Need help buying a laptop

Well you are not going to get an ultrabook like the x1 for $500, are also not going to get a true quad core (an HQ) serries for $500 either.

Your budget is at the get the best special with decent specs in your budget price, cant really be picky, probably not even going to get a PC with an SSD in it.

Budget model from Dell or Asus. Get 15.6" size, i5 U cpu, at minimum 8GB of ram.
You are also a little late for "back to school" specials.
I would frankly check dell-outlet for returned/refurbished models to be able to stretch that budget a little farther.

If you expect the laptop to last thru 4 years of college, your budget is a bit short. $850 is about the entry point here ... My oldest son went away to college with a $1,750 unit .... it was getting a bit tired by the middle of the 3rd year for entertainment purposes. It died in summer break before his 4th year began. It was slow sometimes with schoolwork, the last year of usage but not more than a minor annoyance as only certain programs were affected. In his 4th year he switched to an iPad as it was more conducive to his field of study as a Professional Pilot and the aviation industry if very Apple centric. He now has two of them, that st one is 7 years old and he keeps it in the 2 seater, the newer one is in the 4 seater.

With 3 kids having gone thru college, I spent a lot of time on campus ... visiting the boys and their friends and a Desktop PC was universal (mostly used for gaming / entertainment, but for classes, most kids had MacBooks or iPad's. Unfortunately, given your major, that would not be an appropriate choice. This is what we have been recommending to college students of late:

https/lpc-digital.com/product/sager-np5855-clevo-n855hj/

However, after adding Windows 10, upgrade to 16 GB, and after negotiating a discount from the cash price, you can expect to spend about $800

I would strongly suggest obtaining a list of programs that you will be using during your course of study and then calling the vendors for hardware recommendations. Published system requirements are not very useful as they don't take into account how much longer it takes. Even renting a laptop and trying it out on the programs you might use, would be a wise choice.

BTW, went to the Lenovo site and it says the X1 carbon 'starts at $1400 ... the Zenbooks are at $600 - $1200