Where is the error in this "proof" that 3=0? [duplicate]
Solution 1:
The problem is $x^3=1$ does not imply that $x=1$. The equation $x^3-1=0$ has three possible roots and the root $x=1$ is an additionally generated root.
Solution 2:
Substituting a member of an equation into itself can introduce alien solutions.
E.g. $$x=x^2\implies x^2=x^2.$$
You can do that, provided you keep the initial equation as well.
Safe operations are:
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adding a term to both members;
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multiplying both members by a nonzero factor;
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applying an invertible transformation to both members.
Anything else (e.g. squaring both members) must be done with care.