2 paradoxes

Algebraic manipulation can lead to certain fallacies can give raise to surpising results. Here is a sample of 2 of them taken from the book "one equals zero" by Nitsa Movshovitz-Hadar and John Webb (1998). These paradoxes can be used in class for students to think about.

1) suppose we have the equation x-x^2=1.

Since 0 is not a root of this equation, we can divide both sides by x:

x+1/x=1

So

x-x^2=x+1/x

subtracting both sides by x,

-x^2=1/x

which gives

x=-1

But -1 isn't the answer. What went wrong?

2) Given the equation

log(x-1)^2=2 log 3
2log(x-1)=2log 3
log(x-1)=log 3
x-1=3

so x=4

Is this the only solution? If not? What went wrong in the working above?

Overall, the book proved very interesting for me. There is a variety of paradoxes ranging from algebra to statistics. It also includes calculus and geometry.