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.

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) (Show Source): You can put this solution on YOUR website!
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...
The factor an x from the left side:
Now apply the zero product rule:
or so the two solutions are:
and
Answer by solver91311(24713) (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

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