10[n-6]n=14

Solution for 10[n-6]n=14 equation:

Simplifying
10[n + -6] * n = 14

Reorder the terms:
10[-6 + n] * n = 14

Reorder the terms for easier multiplication:
10n[-6 + n] = 14
[-6 * 10n + n * 10n] = 14
[-60n + 10n2] = 14

Solving
-60n + 10n2 = 14

Solving for variable 'n'.

Reorder the terms:
-14 + -60n + 10n2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + -60n + 10n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-7 + -30n + 5n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-7 + -30n + 5n2)' equal to zero and attempt to solve:

Simplifying
-7 + -30n + 5n2 = 0

Solving
-7 + -30n + 5n2 = 0

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

Divide each side by '5'.
-1.4 + -6n + n2 = 0

Move the constant term to the right:

Add '1.4' to each side of the equation.
-1.4 + -6n + 1.4 + n2 = 0 + 1.4

Reorder the terms:
-1.4 + 1.4 + -6n + n2 = 0 + 1.4

Combine like terms: -1.4 + 1.4 = 0.0
0.0 + -6n + n2 = 0 + 1.4
-6n + n2 = 0 + 1.4

Combine like terms: 0 + 1.4 = 1.4
-6n + n2 = 1.4

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

Add '9' to each side of the equation.
-6n + 9 + n2 = 1.4 + 9

Reorder the terms:
9 + -6n + n2 = 1.4 + 9

Combine like terms: 1.4 + 9 = 10.4
9 + -6n + n2 = 10.4

Factor a perfect square on the left side:
(n + -3)(n + -3) = 10.4

Calculate the square root of the right side: 3.224903099

Break this problem into two subproblems by setting 
(n + -3) equal to 3.224903099 and -3.224903099.

Subproblem 1
n + -3 = 3.224903099

Simplifying
n + -3 = 3.224903099

Reorder the terms:
-3 + n = 3.224903099

Solving
-3 + n = 3.224903099

Solving for variable 'n'.

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

Add '3' to each side of the equation.
-3 + 3 + n = 3.224903099 + 3

Combine like terms: -3 + 3 = 0
0 + n = 3.224903099 + 3
n = 3.224903099 + 3

Combine like terms: 3.224903099 + 3 = 6.224903099
n = 6.224903099

Simplifying
n = 6.224903099

Subproblem 2
n + -3 = -3.224903099

Simplifying
n + -3 = -3.224903099

Reorder the terms:
-3 + n = -3.224903099

Solving
-3 + n = -3.224903099

Solving for variable 'n'.

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

Add '3' to each side of the equation.
-3 + 3 + n = -3.224903099 + 3

Combine like terms: -3 + 3 = 0
0 + n = -3.224903099 + 3
n = -3.224903099 + 3

Combine like terms: -3.224903099 + 3 = -0.224903099
n = -0.224903099

Simplifying
n = -0.224903099

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.224903099, -0.224903099}
Solution
n = {6.224903099, -0.224903099}