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Why can’t I simplify x^2=2x by x

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You said it all : before dividing, you want to make sure that you don’t divide by zero. In your case, by dividing by x, you assume that x cannot be equal to zero. What follows only works if x is not zero. You have to test the other case : what if x equals zero
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You're question has been answered already, but I want to add that it's not really about not dividing by 0, because by doing that step you implicitly assume that x is not equal to 0, it's about losing information. You shouldn't divide by common factors, which may not just be x on its own, unless you know that it leads to solutions that can be scrapped. For example x(x-2) =(x-2). If you were to divide by x-2, you will get x=1, which is a solution, but you will lose the solution x=2, because the division implies that x is not equal to 2.

If you were to have an equation (x-x1)(x-x2)...(x-x\_n) = 0 then dividing by any product of (x-x\_i), where 1<=i=<n will lose the solutions x=x\_i, because the division assumes that x \\neq x\_i. In fact if you were to divide by all of the factors then you will get 0=0, which is correct which means your steps are correct, but you've extracted no new information.
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Look at your second equation. Try to plot it. The curve will cross x=0 at two places. Try to plot all your equations with geogebra, or even better draw them yourself to build an intuitive understanding of what the equation actually means. Im not sure if this directly answered your question the way you wanted to.
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There is never a good reason to divide a polynomial equation by a power of x. You will always be dividing by 0.

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