SOLUTION: If {{{x}}}{{{""=""}}}{{{y + sqrt(y^2+1)}}}, then express y in terms of x

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Question 828275: If x%22%22=%22%22y+%2B+sqrt%28y%5E2%2B1%29, then express y in terms of x
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
x%22%22=%22%22y+%2B+sqrt%28y%5E2%2B1%29

Isolate the square root term:

x-y%22%22=%22%22sqrt%28y%5E2%2B1%29

Square both sides:

%28x-y%29%5E2%22%22=%22%22%28sqrt%28y%5E2%2B1%29%29%5E2

Squaring a square root takes away the square root

%28x-y%29%5E2%22%22=%22%22y%5E2%2B1

Write the left sides as %28x-y%29%28x-y%29

%28x-y%29%28x-y%29%22%22=%22%22y%5E2%2B1

Use FOIL

x%5E2-2xy%2By%5E2%22%22=%22%22y%5E2%2B1

Subtract y%5E2 from both sides:

x%5E2-2xy%22%22=%22%221

Isolate the term that contains y, which is -2xy.
Since it's negative on the left, we isolate it
on the right:

x%5E2-1%22%22=%22%222xy

Divide both sides by 2x

%28x%5E2-1%29%2F%282x%29%22%22=%22%22%282xy%29%2F%282x%29

%28x%5E2-1%29%2F%282x%29%22%22=%22%22%28cross%282x%29y%29%2F%28cross%282x%29%29

%28x%5E2-1%29%2F%282x%29%22%22=%22%22y

y%22%22=%22%22%28x%5E2-1%29%2F%282x%29

Edwin