Answer
$\{-6,2\sqrt6\}$
Work Step by Step
Taking the positive ($x\geq 0$):
$$x=x^2+x-24$$
$$x^2=24$$
$$x=\pm \sqrt{24}=\pm \sqrt{4\cdot6}=\pm 2\sqrt6$$
But $x\geq 0$, so the solution is $x=2\sqrt 6$.
Checking:
$$|2\sqrt{6}|=(2\sqrt6)^2+2\sqrt6-24$$
$$2\sqrt6=2\sqrt6~~True$$
Thus, $x=2\sqrt6$ is a solution.
Taking the negative ($x<0$):
$$x=-(x^2+x-24)$$
$$x=-x^2-x+24$$
$$2x=-x^2+24$$
$$x^2+2x-24=0$$
$$(x+6)(x-4)=0$$
$$x+6=0$$
$$x=-6$$
$$x-4=0$$
$$x=4$$
But $x<0$, so only $x=-6$ fits.
Checking:
$$|-6|=(-6)^2+(-6)-24$$
$$6=6~True$$
Thus, $x=-6$ is a solution.
Therefore, the solution set is:
$$\{-6,2\sqrt6\}$$
