SOLUTION: In solving the question x^2 =x, a student cancels the x on both sides, giving x=1. Comment on the validity of such manipulation. How do we correct such action if it is wrong.

Algebra ->  Equations -> SOLUTION: In solving the question x^2 =x, a student cancels the x on both sides, giving x=1. Comment on the validity of such manipulation. How do we correct such action if it is wrong.      Log On


   

Question 189450This question is from textbook
: In solving the question x^2 =x, a student cancels the x on both sides, giving x=1. Comment on the validity of such manipulation. How do we correct such action if it is wrong. This question is from textbook

Found 2 solutions by Earlsdon, solver91311:
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2+=+x While canceling (dividing by) an x on both sides seems to be a valid mathematical operation, it ignores the fact that the equation is, in fact, a quadratic equation and will have two solutions (roots or zeros).
The correct procedure is to subtract x from both sides giving you...
x%5E2+-+x+=+0 The factor an x from the left side:
x%2A%28x-1%29+=+0 Now apply the zero product rule:
+x+=+0+ or x-1+=+0 so the two solutions are:
+x+=+0+ and x+=+1

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!




Since this is a 2nd degree equation, we know that it must either have two roots or a single root with a multiplicity of 2. So if x = 1 is a single root, then the factors of the polynomial must be:



which is equal to



and which is different than



Therefore, while one of the roots of



is x = 1, there must be a second root different from 1.

The proper method to solve this problem is to factor the polynomial obtained by putting the equation into standard quadratic form:





hence



or



John