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How do I enter an eighth-note tuplet that begins with a quarter note?
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I have a beat with two eighth note, the second of which is tied to another eight note in the next beat. I want to change the first two eighth notes to an eighth-note triplet beginning with a quarter note, and maintain the tie to the eighth note on the next beat.

You can enter 3:2e for 8-note triplets in the popover, then just add a quarter note.
Jesper
For your second question, get the caret to the position where you want the tuplet to start, ensure that eighths are selected in the left panel (hit 5), then type ; 3 Enter. Then shuffle or extend the notes using Alt-Right and/or Shift-Alt-Right (I’m using Shift-Alt-Right here to extend by the caret, and I have my rhythmic grid resolution set to eighths.

Thank you both so much for the helpful replies. I am now able to do this. I am sure that I will have other struggles in learning Dorico after so many years of working with Finale.
I got this to work, but it took me WAY too long and I’m not really sure how I fixed it.
Can someone tell me to best and most efficient way to enter a triplet that starts with a quarter note? It’s an eight note triplet with one quarter and one eight but using the tool keeps trying to make it a quarter note triplet over 2 beats instead of an eight note triplet over 1 beat.
Use the popover ; 3:2e.
The e defines the basis of the tuplet as eighths. If you only enter 3:2 (or use the buttons in the LH panel) Dorico will use whatever the current note duration is as the basis. In your case that likely quarters.
perfect - thank you!
I think getting to know the popover features is going to be important.