6n^2-12n-84=10

Solution for 6n^2-12n-84=10 equation:

Simplifying
6n2 + -12n + -84 = 10

Reorder the terms:
-84 + -12n + 6n2 = 10

Solving
-84 + -12n + 6n2 = 10

Solving for variable 'n'.

Reorder the terms:
-84 + -10 + -12n + 6n2 = 10 + -10

Combine like terms: -84 + -10 = -94
-94 + -12n + 6n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-94 + -12n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-47 + -6n + 3n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-47 + -6n + 3n2)' equal to zero and attempt to solve:

Simplifying
-47 + -6n + 3n2 = 0

Solving
-47 + -6n + 3n2 = 0

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

Divide each side by '3'.
-15.66666667 + -2n + n2 = 0

Move the constant term to the right:

Add '15.66666667' to each side of the equation.
-15.66666667 + -2n + 15.66666667 + n2 = 0 + 15.66666667

Reorder the terms:
-15.66666667 + 15.66666667 + -2n + n2 = 0 + 15.66666667

Combine like terms: -15.66666667 + 15.66666667 = 0.00000000
0.00000000 + -2n + n2 = 0 + 15.66666667
-2n + n2 = 0 + 15.66666667

Combine like terms: 0 + 15.66666667 = 15.66666667
-2n + n2 = 15.66666667

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 15.66666667 + 1

Reorder the terms:
1 + -2n + n2 = 15.66666667 + 1

Combine like terms: 15.66666667 + 1 = 16.66666667
1 + -2n + n2 = 16.66666667

Factor a perfect square on the left side:
(n + -1)(n + -1) = 16.66666667

Calculate the square root of the right side: 4.082482905

Break this problem into two subproblems by setting 
(n + -1) equal to 4.082482905 and -4.082482905.

Subproblem 1
n + -1 = 4.082482905

Simplifying
n + -1 = 4.082482905

Reorder the terms:
-1 + n = 4.082482905

Solving
-1 + n = 4.082482905

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = 4.082482905 + 1

Combine like terms: -1 + 1 = 0
0 + n = 4.082482905 + 1
n = 4.082482905 + 1

Combine like terms: 4.082482905 + 1 = 5.082482905
n = 5.082482905

Simplifying
n = 5.082482905

Subproblem 2
n + -1 = -4.082482905

Simplifying
n + -1 = -4.082482905

Reorder the terms:
-1 + n = -4.082482905

Solving
-1 + n = -4.082482905

Solving for variable 'n'.

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

Add '1' to each side of the equation.
-1 + 1 + n = -4.082482905 + 1

Combine like terms: -1 + 1 = 0
0 + n = -4.082482905 + 1
n = -4.082482905 + 1

Combine like terms: -4.082482905 + 1 = -3.082482905
n = -3.082482905

Simplifying
n = -3.082482905

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.082482905, -3.082482905}
Solution
n = {5.082482905, -3.082482905}