I"ve been asked this many times and can never quite give a good, clear, concise answer (for beginning algebra students) in plain language. I just searched the web and still couldn"t find a simple-to-understand answer for why squaring both sides gives you extraneous solutions.
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$egingroup$ A much better question is "why don't I get extraneous solutions when I add 5 to both sides of an equation?" In the grand scheme of things, "reversible" steps that don't introduce extraneous solutions are rare and unusual -- it's easy to get the wrong idea from elementary school algebra. $endgroup$
I"ve had to teach this to beginning algebra students (in the context of adult ed), so I thought I would put my two cents in.
When you square an equation the result doesn"t remember what the signs of the numbers were before hand. A squared equation is really two equations put into one, the original equation you wanted to solve and a "buddy" equation that has an extra negative sign. The extraneous solutions are solutions of the corresponding buddy equation.
At this point I generally provide a specific example writing down an equation and its buddy (which has an extra negative sign) one above the other and then draw arrows going from both to the common squared equation.
Sometimes I either lead or wrap up the discussion by talking about different arithmetic operations they have learned and point out that if you know a number was obtained by performing addition, multiplication, division, etc. you can always tell me the original number by reversing the process but when we square a number we have no way of knowing the original sign by looking at the result. Often it is helpful to draw diagrams showing the flow of the arithmetic and make them reverse some aritmetic.
For instance you could say that we got 15 when we multiplied a number by 2 and added 1. Then (not emphasizing symbolic algebra but just the arithmatic) the student should be able to reverse the process by subtracting 1 (which gives 14) and dividing by 2 (which gives 7) obtaining the original number.
If your course is anything like mine their first instinct may be to convert the above inversion process into an equation ("rewrite the sentence as an algebraic equation and solve for the unknown"). I think it is very important to not let them do this, more likely than not they"ll get caught up in trying to "solve for x" and probably forget why we are doing this in the first place. The point of the exercise is to teach them the ideas of invertible and non-invertible operations not to see if they can shuffle letters around on a page.
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This should be wrapped up by giving a similar problem but now involving a non-invertible operation. For instance, 25 was obtained by squaring a number. What was the number? They should be able to recognize that this question is flawed because there are two numbers which square to 25.