2y(3y-4y+6)=4

Solution for 2y(3y-4y+6)=4 equation:

Simplifying
2y(3y + -4y + 6) = 4

Reorder the terms:
2y(6 + 3y + -4y) = 4

Combine like terms: 3y + -4y = -1y
2y(6 + -1y) = 4
(6 * 2y + -1y * 2y) = 4
(12y + -2y2) = 4

Solving
12y + -2y2 = 4

Solving for variable 'y'.

Reorder the terms:
-4 + 12y + -2y2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + 12y + -2y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-2 + 6y + -1y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-2 + 6y + -1y2)' equal to zero and attempt to solve:

Simplifying
-2 + 6y + -1y2 = 0

Solving
-2 + 6y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
2 + -6y + y2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -6y + -2 + y2 = 0 + -2

Reorder the terms:
2 + -2 + -6y + y2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -6y + y2 = 0 + -2
-6y + y2 = 0 + -2

Combine like terms: 0 + -2 = -2
-6y + y2 = -2

The y term is -6y.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6y + 9 + y2 = -2 + 9

Reorder the terms:
9 + -6y + y2 = -2 + 9

Combine like terms: -2 + 9 = 7
9 + -6y + y2 = 7

Factor a perfect square on the left side:
(y + -3)(y + -3) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(y + -3) equal to 2.645751311 and -2.645751311.

Subproblem 1
y + -3 = 2.645751311

Simplifying
y + -3 = 2.645751311

Reorder the terms:
-3 + y = 2.645751311

Solving
-3 + y = 2.645751311

Solving for variable 'y'.

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

Add '3' to each side of the equation.
-3 + 3 + y = 2.645751311 + 3

Combine like terms: -3 + 3 = 0
0 + y = 2.645751311 + 3
y = 2.645751311 + 3

Combine like terms: 2.645751311 + 3 = 5.645751311
y = 5.645751311

Simplifying
y = 5.645751311

Subproblem 2
y + -3 = -2.645751311

Simplifying
y + -3 = -2.645751311

Reorder the terms:
-3 + y = -2.645751311

Solving
-3 + y = -2.645751311

Solving for variable 'y'.

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

Add '3' to each side of the equation.
-3 + 3 + y = -2.645751311 + 3

Combine like terms: -3 + 3 = 0
0 + y = -2.645751311 + 3
y = -2.645751311 + 3

Combine like terms: -2.645751311 + 3 = 0.354248689
y = 0.354248689

Simplifying
y = 0.354248689

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.645751311, 0.354248689}
Solution
y = {5.645751311, 0.354248689}