How many different quaternions $q$ are in a satisfying equation $q^2 = 1$?

Question

In my university I started learning quaternions and don't quite understand them. Could I, please, get some help ?

Answer 1

Hint:

$q=\pm 1$ are obviously solutions, so let's attempt to consider $q\neq \pm 1$ with that property.

We have $0=q^2-1=(q-1)(q+1)$. If $q\neq 1$, then $q-1\neq 0$, hence it has an inverse, call it $q'$. Multiply both sides of the equation on the left with $q'$. What results?