Dearly enforummed,

Woe is me. Up is down, left is right, and apparently $\displaystyle -1 = 2$.

Here's the problem from my old textbook:

Here's the very (un)helpful little hint they give:Discuss the followingfalse "proof":

Suppose $\displaystyle a$ is a solution to $\displaystyle a = 1 + a^2$. Clearly $\displaystyle a$ cannot be 0; hence $\displaystyle 1 = (1 / a) + a$.

Use this expression to substitute for 1 in the first conditional equation:

$\displaystyle a = [(1 / a) + a ] + a^2 $. Therefore,

$\displaystyle 0 = (1 / a) + a^2$, and

$\displaystyle -(1/a) = a^2$; hence

$\displaystyle a^3 = -1$.

We now infer that $\displaystyle a = \sqrt[3]{-1} = -1$.

Substituting this value into our original equation, we have

$\displaystyle -1 = 1 + (-1)^2 = 1 + 1 = 2.$

$\displaystyle

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

$

So, my questions are, in order of importance:

0. How do I make a cube root symbol?

1. What kind of sick person dreams up false proofs?

2. How is the hint supposed to help? If I transform the step $\displaystyle a^3 = -1$ to $\displaystyle a^3 + 1 = 0$, and from there to $\displaystyle (a^2 - a + 1)(a + 1) = 0$, I now have a more complicated expression -- one for which the solution is still -1 or +1, neither of which still play well in $\displaystyle a = 1 + a^2$.

3. If I encountered this kind of problem without this helpful hint, how would I know to use it? It seems like a singular rabbit to pull out of a hat...

4. This is the big one... Following normal algebraic processes, starting from a reasonable assumption, this horrific result, uh, resulted. Is this a failure of algebra, or of the idiot user attempting to make it work? Is there a tell-tale sign anywhere along the steps that something went wrong?

I vaguely remember this problem from as far back as high school, and I think it even made sense at the time -- but that was a rather long time ago from my perspective. May it please the court, I humbly beseech any assistance anyone can muster...

- Bill

Attempting to Relearn Calculus

For Masochism and Glory