0=n^2+4n-89

Solution for 0=n^2+4n-89 equation:

Simplifying
0 = n2 + 4n + -89

Reorder the terms:
0 = -89 + 4n + n2

Solving
0 = -89 + 4n + n2

Solving for variable 'n'.

Combine like terms: 0 + 89 = 89
89 + -4n + -1n2 = -89 + 4n + n2 + 89 + -4n + -1n2

Reorder the terms:
89 + -4n + -1n2 = -89 + 89 + 4n + -4n + n2 + -1n2

Combine like terms: -89 + 89 = 0
89 + -4n + -1n2 = 0 + 4n + -4n + n2 + -1n2
89 + -4n + -1n2 = 4n + -4n + n2 + -1n2

Combine like terms: 4n + -4n = 0
89 + -4n + -1n2 = 0 + n2 + -1n2
89 + -4n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
89 + -4n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-89 + 4n + n2 = 0

Move the constant term to the right:

Add '89' to each side of the equation.
-89 + 4n + 89 + n2 = 0 + 89

Reorder the terms:
-89 + 89 + 4n + n2 = 0 + 89

Combine like terms: -89 + 89 = 0
0 + 4n + n2 = 0 + 89
4n + n2 = 0 + 89

Combine like terms: 0 + 89 = 89
4n + n2 = 89

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = 89 + 4

Reorder the terms:
4 + 4n + n2 = 89 + 4

Combine like terms: 89 + 4 = 93
4 + 4n + n2 = 93

Factor a perfect square on the left side:
(n + 2)(n + 2) = 93

Calculate the square root of the right side: 9.643650761

Break this problem into two subproblems by setting 
(n + 2) equal to 9.643650761 and -9.643650761.

Subproblem 1
n + 2 = 9.643650761

Simplifying
n + 2 = 9.643650761

Reorder the terms:
2 + n = 9.643650761

Solving
2 + n = 9.643650761

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = 9.643650761 + -2

Combine like terms: 2 + -2 = 0
0 + n = 9.643650761 + -2
n = 9.643650761 + -2

Combine like terms: 9.643650761 + -2 = 7.643650761
n = 7.643650761

Simplifying
n = 7.643650761

Subproblem 2
n + 2 = -9.643650761

Simplifying
n + 2 = -9.643650761

Reorder the terms:
2 + n = -9.643650761

Solving
2 + n = -9.643650761

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = -9.643650761 + -2

Combine like terms: 2 + -2 = 0
0 + n = -9.643650761 + -2
n = -9.643650761 + -2

Combine like terms: -9.643650761 + -2 = -11.643650761
n = -11.643650761

Simplifying
n = -11.643650761

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.643650761, -11.643650761}