Originally Posted by
CaptainBlack There is a limited range of $\displaystyle X $ over which this can have a solution. For
instance the inequality cannot hold if $\displaystyle |X| \ge 2.5$ . So plug in $\displaystyle -2,-1,0,1,2$ in
turn for $\displaystyle [X]$ and then solve for $\displaystyle X$ . Keep any solutions for which $\displaystyle [X]$ is
equal to the assumed value used.
As an example we will work the case where we assume $\displaystyle [X]=-2$ .
Then we want solutions of:
$\displaystyle
X^4=2 X^2-2
$
which is a quadratic in $\displaystyle X^2$ , so using the quadratic formula:
$\displaystyle
X^2=1 \pm \sqrt{-1}
$ ,
which is complex and so not an admissible solution (I will assume we want
real solutions).
Now lets try assuming $\displaystyle [X]=-1$ , then:
$\displaystyle
X^2=1 \pm \sqrt{0}=1,
$
and so $\displaystyle X=\pm 1$ are solutions of this equation and $\displaystyle X=-1$ is consistent with
our assumption and so is a solution of the original problem.
Carrying on like this you will find all of the remaining solutions.
RonL