3(6+2y)y=1

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Solution for 3(6+2y)y=1 equation: 


Simplifying
3(6 + 2y) * y = 1

Reorder the terms for easier multiplication:
3y(6 + 2y) = 1
(6 * 3y + 2y * 3y) = 1
(18y + 6y2) = 1

Solving
18y + 6y2 = 1

Solving for variable 'y'.

Reorder the terms:
-1 + 18y + 6y2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 18y + 6y2 = 0

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

Divide each side by '6'.
-0.1666666667 + 3y + y2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 3y + 0.1666666667 + y2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 3y + y2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 3y + y2 = 0 + 0.1666666667
3y + y2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
3y + y2 = 0.1666666667

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 0.1666666667 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 0.1666666667 + 2.25

Combine like terms: 0.1666666667 + 2.25 = 2.4166666667
2.25 + 3y + y2 = 2.4166666667

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 2.4166666667

Calculate the square root of the right side: 1.554563176

Break this problem into two subproblems by setting 
(y + 1.5) equal to 1.554563176 and -1.554563176.


Subproblem 1
y + 1.5 = 1.554563176

Simplifying
y + 1.5 = 1.554563176

Reorder the terms:
1.5 + y = 1.554563176

Solving
1.5 + y = 1.554563176

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 1.554563176 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 1.554563176 + -1.5
y = 1.554563176 + -1.5

Combine like terms: 1.554563176 + -1.5 = 0.054563176
y = 0.054563176

Simplifying
y = 0.054563176


Subproblem 2
y + 1.5 = -1.554563176

Simplifying
y + 1.5 = -1.554563176

Reorder the terms:
1.5 + y = -1.554563176

Solving
1.5 + y = -1.554563176

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -1.554563176 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -1.554563176 + -1.5
y = -1.554563176 + -1.5

Combine like terms: -1.554563176 + -1.5 = -3.054563176
y = -3.054563176

Simplifying
y = -3.054563176


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.054563176, -3.054563176}

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