0=x^2+23x+108

Solution for 0=x^2+23x+108 equation:

Simplifying
0 = x2 + 23x + 108

Reorder the terms:
0 = 108 + 23x + x2

Solving
0 = 108 + 23x + x2

Solving for variable 'x'.

Combine like terms: 0 + -108 = -108
-108 + -23x + -1x2 = 108 + 23x + x2 + -108 + -23x + -1x2

Reorder the terms:
-108 + -23x + -1x2 = 108 + -108 + 23x + -23x + x2 + -1x2

Combine like terms: 108 + -108 = 0
-108 + -23x + -1x2 = 0 + 23x + -23x + x2 + -1x2
-108 + -23x + -1x2 = 23x + -23x + x2 + -1x2

Combine like terms: 23x + -23x = 0
-108 + -23x + -1x2 = 0 + x2 + -1x2
-108 + -23x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-108 + -23x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(108 + 23x + x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(108 + 23x + x2)' equal to zero and attempt to solve:

Simplifying
108 + 23x + x2 = 0

Solving
108 + 23x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-108' to each side of the equation.
108 + 23x + -108 + x2 = 0 + -108

Reorder the terms:
108 + -108 + 23x + x2 = 0 + -108

Combine like terms: 108 + -108 = 0
0 + 23x + x2 = 0 + -108
23x + x2 = 0 + -108

Combine like terms: 0 + -108 = -108
23x + x2 = -108

The x term is 23x.  Take half its coefficient (11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
23x + 132.25 + x2 = -108 + 132.25

Reorder the terms:
132.25 + 23x + x2 = -108 + 132.25

Combine like terms: -108 + 132.25 = 24.25
132.25 + 23x + x2 = 24.25

Factor a perfect square on the left side:
(x + 11.5)(x + 11.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(x + 11.5) equal to 4.924428901 and -4.924428901.

Subproblem 1
x + 11.5 = 4.924428901

Simplifying
x + 11.5 = 4.924428901

Reorder the terms:
11.5 + x = 4.924428901

Solving
11.5 + x = 4.924428901

Solving for variable 'x'.

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

Add '-11.5' to each side of the equation.
11.5 + -11.5 + x = 4.924428901 + -11.5

Combine like terms: 11.5 + -11.5 = 0.0
0.0 + x = 4.924428901 + -11.5
x = 4.924428901 + -11.5

Combine like terms: 4.924428901 + -11.5 = -6.575571099
x = -6.575571099

Simplifying
x = -6.575571099

Subproblem 2
x + 11.5 = -4.924428901

Simplifying
x + 11.5 = -4.924428901

Reorder the terms:
11.5 + x = -4.924428901

Solving
11.5 + x = -4.924428901

Solving for variable 'x'.

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

Add '-11.5' to each side of the equation.
11.5 + -11.5 + x = -4.924428901 + -11.5

Combine like terms: 11.5 + -11.5 = 0.0
0.0 + x = -4.924428901 + -11.5
x = -4.924428901 + -11.5

Combine like terms: -4.924428901 + -11.5 = -16.424428901
x = -16.424428901

Simplifying
x = -16.424428901

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-6.575571099, -16.424428901}
Solution
x = {-6.575571099, -16.424428901}