6n^2+7n=192

Solution for 6n^2+7n=192 equation:

Simplifying
6n2 + 7n = 192

Reorder the terms:
7n + 6n2 = 192

Solving
7n + 6n2 = 192

Solving for variable 'n'.

Reorder the terms:
-192 + 7n + 6n2 = 192 + -192

Combine like terms: 192 + -192 = 0
-192 + 7n + 6n2 = 0

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

Divide each side by '6'.
-32 + 1.166666667n + n2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + 1.166666667n + 32 + n2 = 0 + 32

Reorder the terms:
-32 + 32 + 1.166666667n + n2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + 1.166666667n + n2 = 0 + 32
1.166666667n + n2 = 0 + 32

Combine like terms: 0 + 32 = 32
1.166666667n + n2 = 32

The n term is 1.166666667n.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667n + 0.3402777780 + n2 = 32 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667n + n2 = 32 + 0.3402777780

Combine like terms: 32 + 0.3402777780 = 32.340277778
0.3402777780 + 1.166666667n + n2 = 32.340277778

Factor a perfect square on the left side:
(n + 0.5833333335)(n + 0.5833333335) = 32.340277778

Calculate the square root of the right side: 5.686851306

Break this problem into two subproblems by setting 
(n + 0.5833333335) equal to 5.686851306 and -5.686851306.

Subproblem 1
n + 0.5833333335 = 5.686851306

Simplifying
n + 0.5833333335 = 5.686851306

Reorder the terms:
0.5833333335 + n = 5.686851306

Solving
0.5833333335 + n = 5.686851306

Solving for variable 'n'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + n = 5.686851306 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + n = 5.686851306 + -0.5833333335
n = 5.686851306 + -0.5833333335

Combine like terms: 5.686851306 + -0.5833333335 = 5.1035179725
n = 5.1035179725

Simplifying
n = 5.1035179725

Subproblem 2
n + 0.5833333335 = -5.686851306

Simplifying
n + 0.5833333335 = -5.686851306

Reorder the terms:
0.5833333335 + n = -5.686851306

Solving
0.5833333335 + n = -5.686851306

Solving for variable 'n'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + n = -5.686851306 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + n = -5.686851306 + -0.5833333335
n = -5.686851306 + -0.5833333335

Combine like terms: -5.686851306 + -0.5833333335 = -6.2701846395
n = -6.2701846395

Simplifying
n = -6.2701846395

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.1035179725, -6.2701846395}