We can convert this quartic equation to quadratic by putting (x-5) = t (x-5 is mean of x-3 and x-7)
So we get
$(t+2)^4 +(t-2)^4 = 24832$
$2(t^4 + 6 t^2 (-2)^2 + 16) = 24832$
or $t^4 + 24 t^2 = 12400$
$t^4 + 24 t^2 – 12400 = 0$
or $(t^2 – 100)(t^2 + 124) = 0$
so $t^2$ = 100 or t = + or – 10 or x = -5 or 15
Thanks to MarkFL and kaliprasad for participating and provided the good method with correct answers and kali, I remember you once used the same trick to crack my other challenge problem!