3n^2+6n-16972=0

Solution for 3n^2+6n-16972=0 equation:

Simplifying
3n2 + 6n + -16972 = 0

Reorder the terms:
-16972 + 6n + 3n2 = 0

Solving
-16972 + 6n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
-5657.333333 + 2n + n2 = 0

Move the constant term to the right:

Add '5657.333333' to each side of the equation.
-5657.333333 + 2n + 5657.333333 + n2 = 0 + 5657.333333

Reorder the terms:
-5657.333333 + 5657.333333 + 2n + n2 = 0 + 5657.333333

Combine like terms: -5657.333333 + 5657.333333 = 0.000000
0.000000 + 2n + n2 = 0 + 5657.333333
2n + n2 = 0 + 5657.333333

Combine like terms: 0 + 5657.333333 = 5657.333333
2n + n2 = 5657.333333

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 = 5657.333333 + 1

Reorder the terms:
1 + 2n + n2 = 5657.333333 + 1

Combine like terms: 5657.333333 + 1 = 5658.333333
1 + 2n + n2 = 5658.333333

Factor a perfect square on the left side:
(n + 1)(n + 1) = 5658.333333

Calculate the square root of the right side: 75.221893974

Break this problem into two subproblems by setting 
(n + 1) equal to 75.221893974 and -75.221893974.

Subproblem 1
n + 1 = 75.221893974

Simplifying
n + 1 = 75.221893974

Reorder the terms:
1 + n = 75.221893974

Solving
1 + n = 75.221893974

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 = 75.221893974 + -1

Combine like terms: 1 + -1 = 0
0 + n = 75.221893974 + -1
n = 75.221893974 + -1

Combine like terms: 75.221893974 + -1 = 74.221893974
n = 74.221893974

Simplifying
n = 74.221893974

Subproblem 2
n + 1 = -75.221893974

Simplifying
n + 1 = -75.221893974

Reorder the terms:
1 + n = -75.221893974

Solving
1 + n = -75.221893974

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 = -75.221893974 + -1

Combine like terms: 1 + -1 = 0
0 + n = -75.221893974 + -1
n = -75.221893974 + -1

Combine like terms: -75.221893974 + -1 = -76.221893974
n = -76.221893974

Simplifying
n = -76.221893974

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {74.221893974, -76.221893974}