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

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


Simplifying
3x2y + 6xy + y = 0

Reorder the terms:
6xy + 3x2y + y = 0

Solving
6xy + 3x2y + y = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'y'.
y(6x + 3x2 + 1) = 0


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

Simplifying
y = 0

Solving
y = 0

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

Add '-1y' to each side of the equation.
y + -1y = 0 + -1y
Remove the zero:
0 = -1y

Simplifying
0 = -1y

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 '(6x + 3x2 + 1)' equal to zero and attempt to solve:

Simplifying
6x + 3x2 + 1 = 0

Reorder the terms:
1 + 6x + 3x2 = 0

Solving
1 + 6x + 3x2 = 0

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

Divide each side by '3'.
0.3333333333 + 2x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 2x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 2x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 2x + x2 = 0 + -0.3333333333
2x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
2x + x2 = -0.3333333333

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = -0.3333333333 + 1

Reorder the terms:
1 + 2x + x2 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + 2x + x2 = 0.6666666667

Factor a perfect square on the left side:
(x + 1)(x + 1) = 0.6666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(x + 1) equal to 0.816496581 and -0.816496581.


Subproblem 1
x + 1 = 0.816496581

Simplifying
x + 1 = 0.816496581

Reorder the terms:
1 + x = 0.816496581

Solving
1 + x = 0.816496581

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 0.816496581 + -1

Combine like terms: 1 + -1 = 0
0 + x = 0.816496581 + -1
x = 0.816496581 + -1

Combine like terms: 0.816496581 + -1 = -0.183503419
x = -0.183503419

Simplifying
x = -0.183503419


Subproblem 2
x + 1 = -0.816496581

Simplifying
x + 1 = -0.816496581

Reorder the terms:
1 + x = -0.816496581

Solving
1 + x = -0.816496581

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -0.816496581 + -1

Combine like terms: 1 + -1 = 0
0 + x = -0.816496581 + -1
x = -0.816496581 + -1

Combine like terms: -0.816496581 + -1 = -1.816496581
x = -1.816496581

Simplifying
x = -1.816496581


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.183503419, -1.816496581}
Solution
x = {-0.183503419, -1.816496581}

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