What are common mistakes students make with respect to extraneous solutions?
1 Answer
A couple of thoughts...
Explanation:
These are more guesses than informed opinion, but I would suspect the main error is along the lines of not checking for extraneous solutions in the following two cases:

When solving the original problem has involved squaring it somewhere along the line.

When solving a rational equation and having multipled both sides by some factor (which happens to be zero for one of the roots of the derived equation).
Example 1  Squaring
Given:
#sqrt(x+3) = x3#
Square both sides to get:
#x+3 = x^26x+9#
Subtract
#0 = x^27x+6 = (x1)(x6)#
Hence
Example 2  Rational equation
Given:
#x^2/(x1) = (3x2)/(x1)#
Multiply both sides by
#x^2 = 3x2#
Subtract
#0 = x^23x+2 = (x1)(x2)#
Hence