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: 2y

Explanation:

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

So
y-1/y+sqrt((y-1/y)^2+4)
y-1/y+sqrt(y^2-2+1/y^2+4)
y-1/y+sqrt(y^2+2+1/y^2)
y-1/y+sqrt((y+1/y)^2)
y-1/y+y+1/y
2y

Jun 16, 2018

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

Explanation:

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)