8n(2n+3)=12

Solution for 8n(2n+3)=12 equation:

Simplifying
8n(2n + 3) = 12

Reorder the terms:
8n(3 + 2n) = 12
(3 * 8n + 2n * 8n) = 12
(24n + 16n2) = 12

Solving
24n + 16n2 = 12

Solving for variable 'n'.

Reorder the terms:
-12 + 24n + 16n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 24n + 16n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-3 + 6n + 4n2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-3 + 6n + 4n2)' equal to zero and attempt to solve:

Simplifying
-3 + 6n + 4n2 = 0

Solving
-3 + 6n + 4n2 = 0

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

Divide each side by '4'.
-0.75 + 1.5n + n2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 1.5n + 0.75 + n2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 1.5n + n2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 1.5n + n2 = 0 + 0.75
1.5n + n2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
1.5n + n2 = 0.75

The n term is 1.5n.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5n + 0.5625 + n2 = 0.75 + 0.5625

Reorder the terms:
0.5625 + 1.5n + n2 = 0.75 + 0.5625

Combine like terms: 0.75 + 0.5625 = 1.3125
0.5625 + 1.5n + n2 = 1.3125

Factor a perfect square on the left side:
(n + 0.75)(n + 0.75) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(n + 0.75) equal to 1.145643924 and -1.145643924.

Subproblem 1
n + 0.75 = 1.145643924

Simplifying
n + 0.75 = 1.145643924

Reorder the terms:
0.75 + n = 1.145643924

Solving
0.75 + n = 1.145643924

Solving for variable 'n'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + n = 1.145643924 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + n = 1.145643924 + -0.75
n = 1.145643924 + -0.75

Combine like terms: 1.145643924 + -0.75 = 0.395643924
n = 0.395643924

Simplifying
n = 0.395643924

Subproblem 2
n + 0.75 = -1.145643924

Simplifying
n + 0.75 = -1.145643924

Reorder the terms:
0.75 + n = -1.145643924

Solving
0.75 + n = -1.145643924

Solving for variable 'n'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + n = -1.145643924 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + n = -1.145643924 + -0.75
n = -1.145643924 + -0.75

Combine like terms: -1.145643924 + -0.75 = -1.895643924
n = -1.895643924

Simplifying
n = -1.895643924

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.395643924, -1.895643924}
Solution
n = {0.395643924, -1.895643924}