6y=1+3y^2

Solution for 6y=1+3y^2 equation:

Simplifying
6y = 1 + 3y2

Solving
6y = 1 + 3y2

Solving for variable 'y'.

Reorder the terms:
-1 + 6y + -3y2 = 1 + 3y2 + -1 + -3y2

Reorder the terms:
-1 + 6y + -3y2 = 1 + -1 + 3y2 + -3y2

Combine like terms: 1 + -1 = 0
-1 + 6y + -3y2 = 0 + 3y2 + -3y2
-1 + 6y + -3y2 = 3y2 + -3y2

Combine like terms: 3y2 + -3y2 = 0
-1 + 6y + -3y2 = 0

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

Divide each side by '-3'.
0.3333333333 + -2y + y2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2y + -0.3333333333 + y2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2y + y2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2y + y2 = 0 + -0.3333333333
-2y + y2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2y + y2 = -0.3333333333

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

Add '1' to each side of the equation.
-2y + 1 + y2 = -0.3333333333 + 1

Reorder the terms:
1 + -2y + y2 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + -2y + y2 = 0.6666666667

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

Calculate the square root of the right side: 0.816496581

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

Subproblem 1
y + -1 = 0.816496581

Simplifying
y + -1 = 0.816496581

Reorder the terms:
-1 + y = 0.816496581

Solving
-1 + y = 0.816496581

Solving for variable 'y'.

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

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

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

Combine like terms: 0.816496581 + 1 = 1.816496581
y = 1.816496581

Simplifying
y = 1.816496581

Subproblem 2
y + -1 = -0.816496581

Simplifying
y + -1 = -0.816496581

Reorder the terms:
-1 + y = -0.816496581

Solving
-1 + y = -0.816496581

Solving for variable 'y'.

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

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

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

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

Simplifying
y = 0.183503419

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