(3x^2-6y^2+9xy)-(6xy-3y^2)=

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Solution for (3x^2-6y^2+9xy)-(6xy-3y^2)= equation: 


Simplifying
(3x2 + -6y2 + 9xy) + -1(6xy + -3y2) = 0

Reorder the terms:
(9xy + 3x2 + -6y2) + -1(6xy + -3y2) = 0

Remove parenthesis around (9xy + 3x2 + -6y2)
9xy + 3x2 + -6y2 + -1(6xy + -3y2) = 0
9xy + 3x2 + -6y2 + (6xy * -1 + -3y2 * -1) = 0
9xy + 3x2 + -6y2 + (-6xy + 3y2) = 0

Reorder the terms:
9xy + -6xy + 3x2 + -6y2 + 3y2 = 0

Combine like terms: 9xy + -6xy = 3xy
3xy + 3x2 + -6y2 + 3y2 = 0

Combine like terms: -6y2 + 3y2 = -3y2
3xy + 3x2 + -3y2 = 0

Solving
3xy + 3x2 + -3y2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(xy + x2 + -1y2) = 0

Ignore the factor 3.

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

Simplifying
xy + x2 + -1y2 = 0

Solving
xy + x2 + -1y2 = 0

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}
Solution
x = {0.618033989y, -1.618033989y}

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