4y(y+3)=6

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

Simplifying
4y(y + 3) = 6

Reorder the terms:
4y(3 + y) = 6
(3 * 4y + y * 4y) = 6
(12y + 4y2) = 6

Solving
12y + 4y2 = 6

Solving for variable 'y'.

Reorder the terms:
-6 + 12y + 4y2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 12y + 4y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + 6y + 2y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + 6y + 2y2)' equal to zero and attempt to solve:

Simplifying
-3 + 6y + 2y2 = 0

Solving
-3 + 6y + 2y2 = 0

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

Divide each side by '2'.
-1.5 + 3y + y2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 3y + 1.5 + y2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 3y + y2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 3y + y2 = 0 + 1.5
3y + y2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
3y + y2 = 1.5

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 = 1.5 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 1.5 + 2.25

Combine like terms: 1.5 + 2.25 = 3.75
2.25 + 3y + y2 = 3.75

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(y + 1.5) equal to 1.936491673 and -1.936491673.

Subproblem 1
y + 1.5 = 1.936491673

Simplifying
y + 1.5 = 1.936491673

Reorder the terms:
1.5 + y = 1.936491673

Solving
1.5 + y = 1.936491673

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 = 1.936491673 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 1.936491673 + -1.5
y = 1.936491673 + -1.5

Combine like terms: 1.936491673 + -1.5 = 0.436491673
y = 0.436491673

Simplifying
y = 0.436491673

Subproblem 2
y + 1.5 = -1.936491673

Simplifying
y + 1.5 = -1.936491673

Reorder the terms:
1.5 + y = -1.936491673

Solving
1.5 + y = -1.936491673

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 = -1.936491673 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -1.936491673 + -1.5
y = -1.936491673 + -1.5

Combine like terms: -1.936491673 + -1.5 = -3.436491673
y = -3.436491673

Simplifying
y = -3.436491673

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.436491673, -3.436491673}
Solution
y = {0.436491673, -3.436491673}