7y(6y+3)=12

Solution for 7y(6y+3)=12 equation:

Simplifying
7y(6y + 3) = 12

Reorder the terms:
7y(3 + 6y) = 12
(3 * 7y + 6y * 7y) = 12
(21y + 42y2) = 12

Solving
21y + 42y2 = 12

Solving for variable 'y'.

Reorder the terms:
-12 + 21y + 42y2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 21y + 42y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-4 + 7y + 14y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-4 + 7y + 14y2)' equal to zero and attempt to solve:

Simplifying
-4 + 7y + 14y2 = 0

Solving
-4 + 7y + 14y2 = 0

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

Divide each side by '14'.
-0.2857142857 + 0.5y + y2 = 0

Move the constant term to the right:

Add '0.2857142857' to each side of the equation.
-0.2857142857 + 0.5y + 0.2857142857 + y2 = 0 + 0.2857142857

Reorder the terms:
-0.2857142857 + 0.2857142857 + 0.5y + y2 = 0 + 0.2857142857

Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000
0.0000000000 + 0.5y + y2 = 0 + 0.2857142857
0.5y + y2 = 0 + 0.2857142857

Combine like terms: 0 + 0.2857142857 = 0.2857142857
0.5y + y2 = 0.2857142857

The y term is 0.5y.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5y + 0.0625 + y2 = 0.2857142857 + 0.0625

Reorder the terms:
0.0625 + 0.5y + y2 = 0.2857142857 + 0.0625

Combine like terms: 0.2857142857 + 0.0625 = 0.3482142857
0.0625 + 0.5y + y2 = 0.3482142857

Factor a perfect square on the left side:
(y + 0.25)(y + 0.25) = 0.3482142857

Calculate the square root of the right side: 0.590096844

Break this problem into two subproblems by setting 
(y + 0.25) equal to 0.590096844 and -0.590096844.

Subproblem 1
y + 0.25 = 0.590096844

Simplifying
y + 0.25 = 0.590096844

Reorder the terms:
0.25 + y = 0.590096844

Solving
0.25 + y = 0.590096844

Solving for variable 'y'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + y = 0.590096844 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + y = 0.590096844 + -0.25
y = 0.590096844 + -0.25

Combine like terms: 0.590096844 + -0.25 = 0.340096844
y = 0.340096844

Simplifying
y = 0.340096844

Subproblem 2
y + 0.25 = -0.590096844

Simplifying
y + 0.25 = -0.590096844

Reorder the terms:
0.25 + y = -0.590096844

Solving
0.25 + y = -0.590096844

Solving for variable 'y'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + y = -0.590096844 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + y = -0.590096844 + -0.25
y = -0.590096844 + -0.25

Combine like terms: -0.590096844 + -0.25 = -0.840096844
y = -0.840096844

Simplifying
y = -0.840096844

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.340096844, -0.840096844}
Solution
y = {0.340096844, -0.840096844}