6x^3y+6x^2y^2-108xy^3=0

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Solution for 6x^3y+6x^2y^2-108xy^3=0 equation: 


Simplifying
6x3y + 6x2y2 + -108xy3 = 0

Reorder the terms:
-108xy3 + 6x2y2 + 6x3y = 0

Solving
-108xy3 + 6x2y2 + 6x3y = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6xy'.
6xy(-18y2 + xy + x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor 'xy' equal to zero and attempt to solve:

Simplifying
xy = 0

Solving
xy = 0

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

Simplifying
xy = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.

Subproblem 2
Set the factor '(-18y2 + xy + x2)' equal to zero and attempt to solve:

Simplifying
-18y2 + xy + x2 = 0

Reorder the terms:
xy + x2 + -18y2 = 0

Solving
xy + x2 + -18y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '18y2' to each side of the equation.
xy + x2 + -18y2 + 18y2 = 0 + 18y2

Combine like terms: -18y2 + 18y2 = 0
xy + x2 + 0 = 0 + 18y2
xy + x2 = 0 + 18y2
Remove the zero:
xy + x2 = 18y2

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 = 18y2 + 0.25y2

Combine like terms: 18y2 + 0.25y2 = 18.25y2
xy + x2 + 0.25y2 = 18.25y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 18.25y2

Calculate the square root of the right side: 4.272001873y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 4.272001873y and -4.272001873y.


Subproblem 1
x + 0.5y = 4.272001873y

Simplifying
x + 0.5y = 4.272001873y

Solving
x + 0.5y = 4.272001873y

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 = 4.272001873y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 4.272001873y + -0.5y
x = 4.272001873y + -0.5y

Combine like terms: 4.272001873y + -0.5y = 3.772001873y
x = 3.772001873y

Simplifying
x = 3.772001873y


Subproblem 2
x + 0.5y = -4.272001873y

Simplifying
x + 0.5y = -4.272001873y

Solving
x + 0.5y = -4.272001873y

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 = -4.272001873y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -4.272001873y + -0.5y
x = -4.272001873y + -0.5y

Combine like terms: -4.272001873y + -0.5y = -4.772001873y
x = -4.772001873y

Simplifying
x = -4.772001873y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.772001873y, -4.772001873y}
Solution
x = {3.772001873y, -4.772001873y}

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