Lesson 20: Potential れる・える and できる
The potential form exists so that people can say that something "can be done". For five-form verbs, the u is replaced with an e, then added る. So "u" -> "eru". For one-form verbs, it's identical to the passive. Do you hate one-form verbs yet?
食べられる
食べれる (!)
殺せる
死ねる
ありえる
To explain what's going on here, I have to dump some really annoying information on you, so sorry not sorry. Don't worry if your eyes gloss over trying to read it, that's normal. Just read it and move on.
The long form for one-form verbs, 食べられる is the "correct" way to make a potential one-form verb. However, the short form, 食べれる, shows up all the time colloquially, and has done so for a long time, both because it's more regular with five-form verbs and because the long form is obviously ambiguous.
Unfortunately, if context doesn't make it obvious, gut feeling is the only way to know whether a given appearance of one-form られる is passive or potential or both.
In addition to the one-form's ambiguity problem, every passive form can mean the potential sometimes. It's up to you to figure out whether a given use of the passive means a passive, a passive-potential, or just a potential.
ありえる is irregular and is always potential.
Now we can talk about 出来る できる, a one-form verb that means "made" or "completed". It does double duty as する's potential.
出来る 出来ない 出来た 出来ます
昨日は勉強できなかった Yesterday I couldn't study.
できません I can't. (In english, "can not" and "cannot" are assertive in speech. できません isn't, so the contraction is appropriate despite ます)
私は日本語出来ない (unintelligible dysfluent gagging)