3n^2+6n-63=0

Solution for 3n^2+6n-63=0 equation:

Simplifying
3n2 + 6n + -63 = 0

Reorder the terms:
-63 + 6n + 3n2 = 0

Solving
-63 + 6n + 3n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-21 + 2n + n2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-21 + 2n + n2)' equal to zero and attempt to solve:

Simplifying
-21 + 2n + n2 = 0

Solving
-21 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 2n + 21 + n2 = 0 + 21

Reorder the terms:
-21 + 21 + 2n + n2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 2n + n2 = 0 + 21
2n + n2 = 0 + 21

Combine like terms: 0 + 21 = 21
2n + n2 = 21

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 = 21 + 1

Reorder the terms:
1 + 2n + n2 = 21 + 1

Combine like terms: 21 + 1 = 22
1 + 2n + n2 = 22

Factor a perfect square on the left side:
(n + 1)(n + 1) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(n + 1) equal to 4.69041576 and -4.69041576.

Subproblem 1
n + 1 = 4.69041576

Simplifying
n + 1 = 4.69041576

Reorder the terms:
1 + n = 4.69041576

Solving
1 + n = 4.69041576

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 = 4.69041576 + -1

Combine like terms: 1 + -1 = 0
0 + n = 4.69041576 + -1
n = 4.69041576 + -1

Combine like terms: 4.69041576 + -1 = 3.69041576
n = 3.69041576

Simplifying
n = 3.69041576

Subproblem 2
n + 1 = -4.69041576

Simplifying
n + 1 = -4.69041576

Reorder the terms:
1 + n = -4.69041576

Solving
1 + n = -4.69041576

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 = -4.69041576 + -1

Combine like terms: 1 + -1 = 0
0 + n = -4.69041576 + -1
n = -4.69041576 + -1

Combine like terms: -4.69041576 + -1 = -5.69041576
n = -5.69041576

Simplifying
n = -5.69041576

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.69041576, -5.69041576}
Solution
n = {3.69041576, -5.69041576}