How to approach my PhD advisor after failing a qualifiying exam? I'm a first-year Ph.D. student in STEM who recently took their qualifying exams, in which we take two.  I absolutely failed one and passed the other with flying colors (according to a faculty member).  Luckily, in my department, I only have to take the one I did not pass. In terms of why I failed, it was a combination of choking and being deficient in one course which I thought could be remedied by self-studying, however, I did not put in the time. In that regard, I absolutely take the blame.
I am stressed about the future because I believe I embarrassed my advisor* by performing quite poorly. As such, he may have revaluated his opinion of me, which could lead to possible termination. After getting the results, he has not spoken to me about what happened.  Without trying to come off as a narcissist, I was recruited by him given my strong research background and graduate coursework from my bachelor's, which allowed me to start doing research the first semester of my doctoral program.  As such, from the very start, I believe there was a lot of pressure to pass the exams with ease.
Given that I am unsure what he is thinking, would it be appropriate to contact him to clear the air?  I assumed he would bring it up, but I can now see two things happening: We don't speak of it or he will approach me at a later time.  This summer interim it was planned I would continue with research.
TLDR: I failed one of two qualifying exams and am unsure of my future with my advisor given that I have already started research.  Should I approach him about it, and how should I continue forward?
*I apologize if this concern comes off as silly, however, I was raised in a household where honor, integrity, and face are extremely important.
You should talk to your advisor.
It's completely natural to feel embarrassed if you have done badly in an exam where everyone was expecting you to do well. It's also natural to feel you have let your advisor down, especially as it sounds in your case they recruited you specifically.
However, from what you have written, I'd say you have more positives than negatives to take away from the situation: you have started a PhD during a global pandemic and passed half your exams with flying colours. You note yourself that you only failed the other exam because you did not study one topic enough -- that shows great maturity and awareness of what can be done to rectify the situation.
Furthermore, everyone fails exams (or does worse than they were hoping or expecting to). I'd wager you're not the first of your advisor's students to have failed an exam -- or even the tenth. Any advisor worth their salt will not take this personally -- instead, their reaction should be &quot;how can I help my student to succeed when they retake the exam?&quot;. If their reaction is to drop you as a student due to one failed exam, I'd say you're better off without them, frankly.
So, I counsel you to talk to your advisor, stating what you have done here. You can write it in a email first if that feels more comfortable. State what you plan to do to ensure you pass the exam next time around and ask if they have any advice. You can also ask if you should continue to focus on research over the summer or switch to studying for the retake (this might depend on when you will resit the exam -- is it before the summer?).
The key to any successful collaboration in research, and especially a successful student-supervisor relationship, is communication. Keep an open dialogue from the beginning and you will hopefully not run into problems further down the line.
Don't be afraid to ask your advisor for help: that is what they are there for.