If x = y - 1/y and y > 0, simplify, in terms of y: x + square root(x^2 + 4) ?

2 Answers
Jun 16, 2018

Substitute in and cancel out: 2y2y

Explanation:

x+sqrt(x^2+4)x+x2+4
x=y-1/yx=y1y

So
y-1/y+sqrt((y-1/y)^2+4)y1y+(y1y)2+4
y-1/y+sqrt(y^2-2+1/y^2+4)y1y+y22+1y2+4
y-1/y+sqrt(y^2+2+1/y^2)y1y+y2+2+1y2
y-1/y+sqrt((y+1/y)^2)y1y+(y+1y)2
y-1/y+y+1/yy1y+y+1y
2y2y

Jun 16, 2018

x+sqrt(x^2+4)=color(red)(2y)color(white)("xxxxxx")x+x2+4=2yxxxxxx...or at least that's what I got

Explanation:

If x=color(lime)(y-1/y)x=y1y (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=(y1y)+(y1y)2+4

color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+sqrt((color(blue)(y^2 - 2 * y * 1/y+1/(y^2)))+4)XXXXXXXXXXX=y1y+(y22y1y+1y2)+4

color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y) +sqrt(color(brown)((y^2+2+1/(y^2)))XXXXXXXXXXX=y1y+(y2+2+1y2)

color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+sqrt(color(brown)((y+1/y)^2))XXXXXXXXXXX=y1y+(y+1y)2

color(white)("XXXXXXXXXXX")=color(magenta)(y-1/y)+color(brown)(y+1/y)XXXXXXXXXXX=y1y+y+1y

color(white)("XXXXXXXXXXX")=color(red)(2y)XXXXXXXXXXX=2y