2014/114) solve the equation $(x-3)^4 + (x-7)^4 = 24832$

Before I provide the solution I would like to mention that generally a quartic polynomial is not easy to solve but this type of equation

$(x-a)^4 + (x-b)^4= c$ can be converted to a quadratic equation by transformation of

$y= \dfrac{(x-a) + (x-b)}{2}$

as below

we shall put

$y= \dfrac{(x-3) + (x-7)}{2}$ = x- 5

so we get
$(y+2)^4 + (y-2)^4 = 24832$
or $2(y^4 + 6 y^2 (-2)^2 + 16) =  24832$
or $y^4 + 24 y^2 = 12400$
$y^4 + 24 y^2 – 12400 = 0$
 or $(y^2 – 100)(y^2 + 124)  = 0$
so $y^2 = 100$
so $y = \pm 10$

or x = -5 or 15