What is the answer to solvefor y,(x+sqrt(x^2+1))(y+sqrt(y^2+1))=1 ?

The answer to solvefor y,(x+sqrt(x^2+1))(y+sqrt(y^2+1))=1 is y=x {x=0},y=-x

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solvefor y,(x+sqrt(x^2+1))(y+sqrt(y^2+1))=1

Solution

solve for

y, (x + x^{2} + 1) (y + y^{2} + 1) = 1

Solution

y = x {x = 0}, y = - x

Solution steps

(x + x^{2} + 1) (y + y^{2} + 1) = 1

Expand

(x + x^{2} + 1) (y + y^{2} + 1) : y x + y^{2} + 1 x + y x^{2} + 1 + y^{2} + 1 x^{2} + 1

y x + y^{2} + 1 x + y x^{2} + 1 + y^{2} + 1 x^{2} + 1 = 1

Factor

y x + y^{2} + 1 x + y x^{2} + 1 + y^{2} + 1 x^{2} + 1 : (x^{2} + 1 + x) (y + y^{2} + 1)

(x^{2} + 1 + x) (y + y^{2} + 1) = 1

Square both sides:

4 y^{2} x^{2} + 4 y^{2} x x^{2} + 1 + 2 y^{2} + 4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x^{2} + 2 x x^{2} + 1 + 1 = 1

4 y^{2} x^{2} + 4 y^{2} x x^{2} + 1 + 2 y^{2} + 4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x^{2} + 2 x x^{2} + 1 + 1 = 1

Subtract

1

from both sides

4 y^{2} x^{2} + 4 y^{2} x x^{2} + 1 + 2 y^{2} + 4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x^{2} + 2 x x^{2} + 1 + 1 - 1 = 1 - 1

Simplify

4 y^{2} x^{2} + 4 y^{2} x x^{2} + 1 + 2 y^{2} + 4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x^{2} + 2 x x^{2} + 1 = 0

Move

4 y^{2} x^{2}

to the right side

4 y^{2} x x^{2} + 1 + 2 y^{2} + 4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x^{2} + 2 x x^{2} + 1 = - 4 x^{2} y^{2}

Move

4 y^{2} x x^{2} + 1

to the right side

2 y^{2} + 4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x^{2} + 2 x x^{2} + 1 = - 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1

Move

2 y^{2}

to the right side

4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x^{2} + 2 x x^{2} + 1 = - 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1 - 2 y^{2}

Move

2 x^{2}

to the right side

4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 2 x x^{2} + 1 = - 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1 - 2 y^{2} - 2 x^{2}

Move

2 x x^{2} + 1

to the right side

4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 = - 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1 - 2 y^{2} - 2 x^{2} - 2 x x^{2} + 1

Subtract

- 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1 - 2 y^{2} - 2 x^{2} - 2 x x^{2} + 1

from both sides

4 y x^{2} y^{2} + 1 + 4 y x y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 - (- 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1 - 2 y^{2} - 2 x^{2} - 2 x x^{2} + 1) = - 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1 - 2 y^{2} - 2 x^{2} - 2 x x^{2} + 1 - (- 4 x^{2} y^{2} - 4 y^{2} x x^{2} + 1 - 2 y^{2} - 2 x^{2} - 2 x x^{2} + 1)

Simplify

4 x^{2} y y^{2} + 1 + 4 x y y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 4 x^{2} y^{2} + 4 x y^{2} x^{2} + 1 + 2 y^{2} + 2 x^{2} + 2 x x^{2} + 1 = 0

Factor

4 x^{2} y y^{2} + 1 + 4 x y y^{2} + 1 x^{2} + 1 + 2 y y^{2} + 1 + 4 x^{2} y^{2} + 4 x y^{2} x^{2} + 1 + 2 y^{2} + 2 x^{2} + 2 x x^{2} + 1 : 2 (2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 + 2 x^{2} y^{2} + 2 x y^{2} x^{2} + 1 + y^{2} + x^{2} + x x^{2} + 1)

2 (2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 + 2 x^{2} y^{2} + 2 x y^{2} x^{2} + 1 + y^{2} + x^{2} + x x^{2} + 1) = 0

Divide both sides by

2

2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 + 2 x^{2} y^{2} + 2 x y^{2} x^{2} + 1 + y^{2} + x^{2} + x x^{2} + 1 = 0

Move

2 x^{2} y^{2}

to the right side

2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 + 2 x y^{2} x^{2} + 1 + y^{2} + x^{2} + x x^{2} + 1 = - 2 x^{2} y^{2}

Move

2 x y^{2} x^{2} + 1

to the right side

2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 + y^{2} + x^{2} + x x^{2} + 1 = - 2 x^{2} y^{2} - 2 x y^{2} x^{2} + 1

Move

y^{2}

to the right side

2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 + x^{2} + x x^{2} + 1 = - 2 x^{2} y^{2} - 2 x y^{2} x^{2} + 1 - y^{2}

Move

x^{2}

to the right side

2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 + x x^{2} + 1 = - 2 x^{2} y^{2} - 2 x y^{2} x^{2} + 1 - y^{2} - x^{2}

Move

x x^{2} + 1

to the right side

2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 = - 2 x^{2} y^{2} - 2 x y^{2} x^{2} + 1 - y^{2} - x^{2} - x x^{2} + 1

Factor

2 x^{2} y y^{2} + 1 + 2 x y y^{2} + 1 x^{2} + 1 + y y^{2} + 1 : y y^{2} + 1 (2 x^{2} + 2 x x^{2} + 1 + 1)

y y^{2} + 1 (2 x^{2} + 2 x x^{2} + 1 + 1) = - 2 x^{2} y^{2} - 2 x y^{2} x^{2} + 1 - y^{2} - x^{2} - x x^{2} + 1

Divide both sides by

y (2 x^{2} + 2 x x^{2} + 1 + 1)

y^{2} + 1 = \frac{- 2 x ^{2} y ^{2} - 2 x y ^{2} x ^{2} + 1 - y ^{2} - x ^{2} - x x ^{2} + 1}{y ( 1 + 2 x ^{2} + 2 x x ^{2} + 1 )}

Square both sides:

y^{2} + 1 = \frac{8 y ^{4} x ^{4} + 8 y ^{2} x ^{4} + 2 x ^{4} + 8 y ^{4} x ^{3} x ^{2} + 1 + 8 y ^{2} x ^{3} x ^{2} + 1 + 2 x ^{3} x ^{2} + 1 + 8 y ^{4} x ^{2} + 6 y ^{2} x ^{2} + x ^{2} + 4 y ^{4} x x ^{2} + 1 + 2 y ^{2} x x ^{2} + 1 + y ^{4}}{8 y ^{2} x ^{4} + 8 y ^{2} x ^{2} + 4 y ^{2} x x ^{2} + 1 + 8 y ^{2} x ^{3} x ^{2} + 1 + y ^{2}}

y^{2} + 1 = \frac{8 y ^{4} x ^{4} + 8 y ^{2} x ^{4} + 2 x ^{4} + 8 y ^{4} x ^{3} x ^{2} + 1 + 8 y ^{2} x ^{3} x ^{2} + 1 + 2 x ^{3} x ^{2} + 1 + 8 y ^{4} x ^{2} + 6 y ^{2} x ^{2} + x ^{2} + 4 y ^{4} x x ^{2} + 1 + 2 y ^{2} x x ^{2} + 1 + y ^{4}}{8 y ^{2} x ^{4} + 8 y ^{2} x ^{2} + 4 y ^{2} x x ^{2} + 1 + 8 y ^{2} x ^{3} x ^{2} + 1 + y ^{2}}

Solve

y^{2} + 1 = \frac{8 y ^{4} x ^{4} + 8 y ^{2} x ^{4} + 2 x ^{4} + 8 y ^{4} x ^{3} x ^{2} + 1 + 8 y ^{2} x ^{3} x ^{2} + 1 + 2 x ^{3} x ^{2} + 1 + 8 y ^{4} x ^{2} + 6 y ^{2} x ^{2} + x ^{2} + 4 y ^{4} x x ^{2} + 1 + 2 y ^{2} x x ^{2} + 1 + y ^{4}}{8 y ^{2} x ^{4} + 8 y ^{2} x ^{2} + 4 y ^{2} x x ^{2} + 1 + 8 y ^{2} x ^{3} x ^{2} + 1 + y ^{2}} : y = x, y = - x

y = x, y = - x

Verify Solutions:

y = x {x = 0}, y = - x

True

The solutions are

y = x {x = 0}, y = - x

Graph

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Frequently Asked Questions (FAQ)

What is the answer to solvefor y,(x+sqrt(x^2+1))(y+sqrt(y^2+1))=1 ?
The answer to solvefor y,(x+sqrt(x^2+1))(y+sqrt(y^2+1))=1 is y=x {x=0},y=-x