If #x=color(lime)(y-1/y)# (with #y > 0#)
then
#color(white)("XXX")x+sqrt(x^2+4)=(color(lime)(y-1/y))+sqrt((color(lime)(y-1/y))^2+4)#
#color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+sqrt((color(blue)(y^2 - 2 * y * 1/y+1/(y^2)))+4)#
#color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y) +sqrt(color(brown)((y^2+2+1/(y^2)))#
#color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+sqrt(color(brown)((y+1/y)^2))#
#color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+color(brown)(y+1/y)#
#color(white)("XXXXXXXXXXX")=color(red)(2y)#