2y(3y+6)=41

Solution for 2y(3y+6)=41 equation:

Simplifying
2y(3y + 6) = 41

Reorder the terms:
2y(6 + 3y) = 41
(6 * 2y + 3y * 2y) = 41
(12y + 6y2) = 41

Solving
12y + 6y2 = 41

Solving for variable 'y'.

Reorder the terms:
-41 + 12y + 6y2 = 41 + -41

Combine like terms: 41 + -41 = 0
-41 + 12y + 6y2 = 0

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

Divide each side by '6'.
-6.833333333 + 2y + y2 = 0

Move the constant term to the right:

Add '6.833333333' to each side of the equation.
-6.833333333 + 2y + 6.833333333 + y2 = 0 + 6.833333333

Reorder the terms:
-6.833333333 + 6.833333333 + 2y + y2 = 0 + 6.833333333

Combine like terms: -6.833333333 + 6.833333333 = 0.000000000
0.000000000 + 2y + y2 = 0 + 6.833333333
2y + y2 = 0 + 6.833333333

Combine like terms: 0 + 6.833333333 = 6.833333333
2y + y2 = 6.833333333

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 = 6.833333333 + 1

Reorder the terms:
1 + 2y + y2 = 6.833333333 + 1

Combine like terms: 6.833333333 + 1 = 7.833333333
1 + 2y + y2 = 7.833333333

Factor a perfect square on the left side:
(y + 1)(y + 1) = 7.833333333

Calculate the square root of the right side: 2.798809271

Break this problem into two subproblems by setting 
(y + 1) equal to 2.798809271 and -2.798809271.

Subproblem 1
y + 1 = 2.798809271

Simplifying
y + 1 = 2.798809271

Reorder the terms:
1 + y = 2.798809271

Solving
1 + y = 2.798809271

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 = 2.798809271 + -1

Combine like terms: 1 + -1 = 0
0 + y = 2.798809271 + -1
y = 2.798809271 + -1

Combine like terms: 2.798809271 + -1 = 1.798809271
y = 1.798809271

Simplifying
y = 1.798809271

Subproblem 2
y + 1 = -2.798809271

Simplifying
y + 1 = -2.798809271

Reorder the terms:
1 + y = -2.798809271

Solving
1 + y = -2.798809271

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 = -2.798809271 + -1

Combine like terms: 1 + -1 = 0
0 + y = -2.798809271 + -1
y = -2.798809271 + -1

Combine like terms: -2.798809271 + -1 = -3.798809271
y = -3.798809271

Simplifying
y = -3.798809271

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.798809271, -3.798809271}