If x=color(lime)(y-1/y)x=y−1y (with y > 0y>0)
then
color(white)("XXX")x+sqrt(x^2+4)=(color(lime)(y-1/y))+sqrt((color(lime)(y-1/y))^2+4)XXXx+√x2+4=(y−1y)+√(y−1y)2+4
color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+sqrt((color(blue)(y^2 - 2 * y * 1/y+1/(y^2)))+4)XXXXXXXXXXX=y−1y+√(y2−2⋅y⋅1y+1y2)+4
color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y) +sqrt(color(brown)((y^2+2+1/(y^2)))XXXXXXXXXXX=y−1y+√(y2+2+1y2)
color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+sqrt(color(brown)((y+1/y)^2))XXXXXXXXXXX=y−1y+√(y+1y)2
color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+color(brown)(y+1/y)XXXXXXXXXXX=y−1y+y+1y
color(white)("XXXXXXXXXXX")=color(red)(2y)XXXXXXXXXXX=2y