a = 1+a^2. Since a can't equal zero, (then 0 = 1), I can divide by a, yielding
1=1/a + a. Substituting for 1 in the original equation,
a=1/a + a + a^2
-1/a = a^2
-1=a^3
a=-1
Checking into the original equation,
-1 = 1+(-1)^2
-1=2
Clearly, -1 doesn't equal two- however, can you find the error?
1 comment:
Okay, fine. Here's why it's wrong.
When I give the solution of a^3=-1 as a=-1, I have ignored the fact that there are two other solutions to this equation. (They can be derived easily from the original quadratic equation using the quadratic formula.) Because of a careful manipulation of variables and the multiplication by a, I have added the solution a=-1; however, it isn't a true solution because it wasn't implied by the original equation.
More of these will come when I think of them!
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