Solving for Vt and Id for a depletion-type NMOS transistor

1. The problem statement, all variables and given/known data
Here is the problem:

Here is the solution from the professor:

So for this problem there are two cases that we can use to solve for Vt. This is what confuses me because you can easily for for Vt in each case and end up with 4 Vt values. But for some reason, the solution suggests that you can combine the two equations and solve for Vt. This results in Vt = -0.5 and -2. However, upon plugging the Vt values back in, for example if you plug Vt = -0.5 into the equation 9 = (1-Vt)2 it will not give you the correct value. Is it possible for solve for all four Vt values and some how rule them out until you get the right Vt?

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