So say that I have the quadratic equation ##x^2 - 2x +1 = 0##. If I multiply each side by ##x##, a valid operation, I get the new equation ##x^3 - 2x^2 + x = 0##, which has a different solution set as the first, namely ##\left\{ {0, 1}\right\}##. If I make the substitution ##x^2 = 2x -1 = 0##...