In remedial algebra, we learn that the graph of $y=(\sqrt x)^2$ is only in the first quadrant. We know this is the correct graph for the equation. This is because we know $y=x$ and $x \ge 0$.
However, a student in a higher level algebra class got the full line where $y=x$, $x$ is all real numbers. Her logic was that if you take $x=-4$, you'd take the square root and get $2i$, and then you'd square that to get $-4$ because $i^2 = -1$ and $2^2 = 4$. This would give you a line that goes into the negatives, which is different from the correct answer we are told, which is a line only in the first quadrant.
This student did not thrive in class, however she made a very interesting argument. She did not see the point of what she came up with and just wanted to pass the class. How would I, as a teacher, try to celebrate this students solution? Although we don’t know if it’s really right or wrong, how could a teacher try and justify her answer and what could a teacher say to encourage her thinking and show a connection to how complex numbers were made?