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:
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When solving the original problem has involved squaring it somewhere along the line.
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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) = x-3#
Square both sides to get:
#x+3 = x^2-6x+9#
Subtract
#0 = x^2-7x+6 = (x-1)(x-6)#
Hence
Example 2 - Rational equation
Given:
#x^2/(x-1) = (3x-2)/(x-1)#
Multiply both sides by
#x^2 = 3x-2#
Subtract
#0 = x^2-3x+2 = (x-1)(x-2)#
Hence