Question

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

Answer 1

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`

Answer 2

`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)`