Cracked egg leaves egg white residue. Should I worry about the other eggs?
If the egg actually exploded as in the title, then no, that egg is probably highly contaminated.  Not only is it bad, but nothing its contents have touched can be considered safe to use and should be discarded.

I assume however that you actually have an egg that was broken in handling.  In that case, only the broken egg is a loss.  Other answers and comments have addressed that, choice by location and your own personal sense of safe for you.  I hate wasting food, but eggs today are a relatively inexpensive commodity for most of us, so if you feel uneasy, error on the side of safety.

When an egg is only partially cracked, but the membrane is intact, that egg is normally safe for use for several days.  You have lost some protection, but not the immediate integrity of the egg.  One issue though is you do not know when it was cracked.  In your possession can be safe, my personal rule being about a week.  But if it already was cracked, I do not know if it happened before washing (in the US) or after.  If before, the egg is compromised.  Now, this is one spot that the EU with non-washed eggs are at a slight, very slight, disadvantage.  IMO, unwashed, a cracked egg is compromised even if the membrane is intact.  That paragraph is not exactly your question, but could be for people searching that find their way here.