Question

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

Answer 1

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)