Say I have the equation: x^2 = -2x
I am tempted to simply divide each side by x, which results in: x = -2
But I understand that to be an incomplete solution. If I simply differently, I wind up with:
x^2 = -2x
x^2 + 2x = 0
x(x + 2) = 0
Hence x = 0, -2.
Can anyone please explain this to me conceptually? How can I avoid repeating this mistake? Should I ALWAYS try to set up exponential equations by putting 0 (zero) on one side and all other terms on the other side, i.e.
POLYNOMIAL = 0
In short the answer is YES, you have to equate a polynomial to a zero, by bringing all the terms to LHS. Solving by bringing all the variables in a polynomial equation to the LHS allows you to extract all the "real" roots of the equation, if they exist. Now, there are equations like x^2+1 which does not have real roots but that's besides the point.
By solving equations the way you are solving you are not guaranteed to find all the roots of the equation. Clearly from you own example, you have seen that by using your method "0" is not seen as one of the roots of the equation.