- #1
Mr Davis 97
- 1,462
- 44
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##, then I now have ##x^3 - 2(2x - 1) + x = x^3 - 3x + 2##, which has the solution set ##\left\{ {-2,1}\right\}##. I could repeat this process ad infinitum, but I am just wondering what is going on. If I am just doing valid operations and making valid substitutions, then why is the solution set constantly changing from what is originally was, i.e, ##\left\{ {1}\right\}##?
Also, given that I transformed ##x^2 - 2x + 1 = 0## to ##x^3 - 3x + 2 = 0##, is there any way to go in the reverse direction, that is, to start with ##x^3 - 3x + 2 = 0## and go to ##x^2 - 2x + 1 = 0##?
Also, given that I transformed ##x^2 - 2x + 1 = 0## to ##x^3 - 3x + 2 = 0##, is there any way to go in the reverse direction, that is, to start with ##x^3 - 3x + 2 = 0## and go to ##x^2 - 2x + 1 = 0##?