A challenging algebra problem from the 2011 AMC 10b

This problem gave my son some trouble this morning:

Problem #19 from the 2011 AMC 10b

The problem asks you to find the product of the roots of the equation:

\sqrt{ 5|x| + 8} = \sqrt{x^2 - 16}

A lot of ideas from Algebra come into play on this problem and we had a good conversation about it tonight.

We started by talking through his ideas about the problem and found a couple of interesting misconceptions about algebra and equality right at the beginning:

Next we went back to look if the 4 solutions that we found to the equation were actually solutions:

Finally, I went back through the problem to show him how the 3 and -3 showed up even though they are not solutions.

The emergence of these spurious solutions makes this a really instructive problem.


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