What are common mistakes students make with respect to extraneous solutions?

1 Answer
Aug 5, 2017

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).

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Example 1 - Squaring

Given:

#sqrt(x+3) = x-3#

Square both sides to get:

#x+3 = x^2-6x+9#

Subtract #x+3# from both sides to get:

#0 = x^2-7x+6 = (x-1)(x-6)#

Hence #x=1# or #x=6" "# (but #x=1# is not a valid solution of the original equation)

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Example 2 - Rational equation

Given:

#x^2/(x-1) = (3x-2)/(x-1)#

Multiply both sides by #(x-1)# to get:

#x^2 = 3x-2#

Subtract #3x-2# from both sides to get:

#0 = x^2-3x+2 = (x-1)(x-2)#

Hence #x=1# or #x=2" "# (but #x=1# is not a valid solution of the original equation)