4x^2+4xy=2y^2

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Solution for 4x^2+4xy=2y^2 equation: 


Simplifying
4x2 + 4xy = 2y2

Reorder the terms:
4xy + 4x2 = 2y2

Solving
4xy + 4x2 = 2y2

Solving for variable 'x'.

Combine like terms: 2y2 + -2y2 = 0
4xy + 4x2 + -2y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(2xy + 2x2 + -1y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(2xy + 2x2 + -1y2)' equal to zero and attempt to solve:

Simplifying
2xy + 2x2 + -1y2 = 0

Solving
2xy + 2x2 + -1y2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
xy + x2 + -0.5y2 = 0

Move the constant term to the right:

Add '0.5y2' to each side of the equation.
xy + x2 + -0.5y2 + 0.5y2 = 0 + 0.5y2

Combine like terms: -0.5y2 + 0.5y2 = 0.0
xy + x2 + 0.0 = 0 + 0.5y2
xy + x2 = 0 + 0.5y2
Remove the zero:
xy + x2 = 0.5y2

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.
y2 + x2 + 0.25y2 = 0.5y2 + 0.25y2

Reorder the terms:
x2 + y2 + 0.25y2 = 0.5y2 + 0.25y2

Combine like terms: y2 + 0.25y2 = 1.25y2
x2 + 1.25y2 = 0.5y2 + 0.25y2

Combine like terms: 0.5y2 + 0.25y2 = 0.75y2
x2 + 1.25y2 = 0.75y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 0.75y2

Calculate the square root of the right side: 0.866025404y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 0.866025404y and -0.866025404y.


Subproblem 1
x + 0.5y = 0.866025404y

Simplifying
x + 0.5y = 0.866025404y

Solving
x + 0.5y = 0.866025404y

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 = 0.866025404y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 0.866025404y + -0.5y
x = 0.866025404y + -0.5y

Combine like terms: 0.866025404y + -0.5y = 0.366025404y
x = 0.366025404y

Simplifying
x = 0.366025404y


Subproblem 2
x + 0.5y = -0.866025404y

Simplifying
x + 0.5y = -0.866025404y

Solving
x + 0.5y = -0.866025404y

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 = -0.866025404y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -0.866025404y + -0.5y
x = -0.866025404y + -0.5y

Combine like terms: -0.866025404y + -0.5y = -1.366025404y
x = -1.366025404y

Simplifying
x = -1.366025404y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.366025404y, -1.366025404y}
Solution
x = {0.366025404y, -1.366025404y}

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