Hear them out

Listen to your patients. You can’t solve a problem you don’t know you have, and that’s why listening is critical.
Oftentimes, the patient just needs to vent – for whatever reason. If you let them, it can diffuse 80% of their anger so they calm down and are now in a rational state of mind to talk and come up with solutions.
Just listen.
You don’t have to agree, dispute or negotiate what they are saying.
Just listen.
Keep your ego in check and squash defensiveness. Although the patient may be attacking your skills or integrity, keep your cool and remember the objective here is to neutralize the unhappy patient so they don’t damage your reputation.
Don’t make excuses, don’t qualify and definitely don’t disparage the patient for speaking their mind. Just listen and learn.
Make a Sincere Apology
This is tricky because you don’t want to admit wrongdoing; however; you do want to acknowledge your sincere concern for your patient’s feelings.
Be honest with yourself. Is the patient being completely irrational or is their room for improvement in the result they got?
The answer is usually somewhere in the middle so apologize for the situation and acknowledge how they feel by saying something like, ““Sara, I’m sorry you’re not happy with (me, staff, result) so let’s talk about a solution.”
Find Out What The Cosmetic Patient Wants
Now it’s time to make it right. Find out what the cosmetic patient wants. Ask a simple question like, “What is it you want me to do for you?”
Did you notice you are NOT admitting guilt here. You are simply acknowledging their feelings.
There’s a good chance it’s going to be easier to appease them than you thought. If it is something reasonable, why not give it to them and move on?
If the unhappy patient wants more than that, like a refund, it’s best to contact your lawyer to draw up an agreement to go with it so they agree not to talk about you online or offline.
More helpful insights about handling cosmetic patients are in my book, Your Aesthetic Practice/What Your Patients Are Saying.