Why can't you square both sides of an equation?

If two things are equal, then so long as you do the same thing to both, they will remain equal. There is nothing wrong with taking the square of both sides of an equation. However, you have to be careful if you want to take the square root of both sides, because the square root is not a normal function: it has two values $\pm \sqrt x$. By convention, the positive square root is chosen, and that is what people mean when they say "the square root". But equations don't care about our conventions. The fact that $(-1)^2 = 1^2$ certainly doesn't imply that $-1 = 1$.

In other words, if $x^2 = y^2$, then taking the square root (using the stated convention) of both sides results in $|x| = |y|$, not in $x=y$.

For these reasons, if you have an equation containing an unknown, then squaring both sides of it can introduce new solutions, so you have to be careful. For instance, the equation $x=1$ obviously has only one solution (namely $x=1$!) but squaring both sides of it yields the equation $x^2=1$ which has the two solutions $x=\pm 1$.

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.

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.

You can.

However, you must be careful that you could have introduced extraneous solutions. Hence, you have to check that your solutions of the squared equation do actually satisfy the original equation.

Case in point: Solve $x -1 = 1$.

If we square both sides, we get $x^2 - 2x + 1 = 1^2$, or that $0 = x^2 - 2x = x(x-2)$. This has solutions $x=0, 2$. We then have to check back if they satisfy the original equation.