2023 AMC 10B Problem 22

How many distinct values of x satisfy \[\lfloor x \rfloor^2 - 3x + 2 = 0,\] where \lfloor x \rfloor denotes the largest integer less than or equal to x?

\textbf{(A)} an infinite number \qquad \textbf{(B)} 4 \qquad \textbf{(C)} 2 \qquad \textbf{(D)} 3 \qquad \textbf{(E)} 0

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.