(x*x)+xy-(2y*2y)=0

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Solution for (x*x)+xy-(2y*2y)=0 equation: 


Simplifying
(x * x) + xy + -1(2y * 2y) = 0

Multiply x * x
(x2) + xy + -1(2y * 2y) = 0
x2 + xy + -1(2y * 2y) = 0

Reorder the terms for easier multiplication:
x2 + xy + -1(2 * 2y * y) = 0

Multiply 2 * 2
x2 + xy + -1(4y * y) = 0

Multiply y * y
x2 + xy + -1(4y2) = 0

Remove parenthesis around (4y2)
x2 + xy + -1 * 4y2 = 0

Multiply -1 * 4
x2 + xy + -4y2 = 0

Reorder the terms:
xy + x2 + -4y2 = 0

Solving
xy + x2 + -4y2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '4y2' to each side of the equation.
xy + x2 + -4y2 + 4y2 = 0 + 4y2

Combine like terms: -4y2 + 4y2 = 0
xy + x2 + 0 = 0 + 4y2
xy + x2 = 0 + 4y2
Remove the zero:
xy + x2 = 4y2

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 = 4y2 + 0.25y2

Combine like terms: 4y2 + 0.25y2 = 4.25y2
xy + x2 + 0.25y2 = 4.25y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 4.25y2

Calculate the square root of the right side: 2.061552813y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 2.061552813y and -2.061552813y.


Subproblem 1
x + 0.5y = 2.061552813y

Simplifying
x + 0.5y = 2.061552813y

Solving
x + 0.5y = 2.061552813y

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 = 2.061552813y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 2.061552813y + -0.5y
x = 2.061552813y + -0.5y

Combine like terms: 2.061552813y + -0.5y = 1.561552813y
x = 1.561552813y

Simplifying
x = 1.561552813y


Subproblem 2
x + 0.5y = -2.061552813y

Simplifying
x + 0.5y = -2.061552813y

Solving
x + 0.5y = -2.061552813y

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 = -2.061552813y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -2.061552813y + -0.5y
x = -2.061552813y + -0.5y

Combine like terms: -2.061552813y + -0.5y = -2.561552813y
x = -2.561552813y

Simplifying
x = -2.561552813y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.561552813y, -2.561552813y}

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