x^2+x*y-y^2=0

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Solution for x^2+x*y-y^2=0 equation: 


Simplifying
x2 + x * y + -1y2 = 0

Multiply x * y
x2 + xy + -1y2 = 0

Reorder the terms:
xy + x2 + -1y2 = 0

Solving
xy + x2 + -1y2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add 'y2' to each side of the equation.
xy + x2 + -1y2 + y2 = 0 + y2

Combine like terms: -1y2 + y2 = 0
xy + x2 + 0 = 0 + y2
xy + x2 = 0 + y2
Remove the zero:
xy + x2 = y2

The x term is xy.  Take half its coefficient (0.5y).
Square it (0.25y2) and add it to both sides.

Add '0.25y2' to each side of the equation.
xy + x2 + 0.25y2 = y2 + 0.25y2

Combine like terms: y2 + 0.25y2 = 1.25y2
xy + x2 + 0.25y2 = 1.25y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 1.25y2

Calculate the square root of the right side: 1.118033989y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 1.118033989y and -1.118033989y.


Subproblem 1
x + 0.5y = 1.118033989y

Simplifying
x + 0.5y = 1.118033989y

Solving
x + 0.5y = 1.118033989y

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5y' to each side of the equation.
x + 0.5y + -0.5y = 1.118033989y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 1.118033989y + -0.5y
x = 1.118033989y + -0.5y

Combine like terms: 1.118033989y + -0.5y = 0.618033989y
x = 0.618033989y

Simplifying
x = 0.618033989y


Subproblem 2
x + 0.5y = -1.118033989y

Simplifying
x + 0.5y = -1.118033989y

Solving
x + 0.5y = -1.118033989y

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5y' to each side of the equation.
x + 0.5y + -0.5y = -1.118033989y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -1.118033989y + -0.5y
x = -1.118033989y + -0.5y

Combine like terms: -1.118033989y + -0.5y = -1.618033989y
x = -1.618033989y

Simplifying
x = -1.618033989y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.618033989y, -1.618033989y}

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