x(x+1)=109

Solution for x(x+1)=109 equation:

Simplifying
x(x + 1) = 109

Reorder the terms:
x(1 + x) = 109
(1 * x + x * x) = 109
(1x + x2) = 109

Solving
1x + x2 = 109

Solving for variable 'x'.

Reorder the terms:
-109 + 1x + x2 = 109 + -109

Combine like terms: 109 + -109 = 0
-109 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '109' to each side of the equation.
-109 + 1x + 109 + x2 = 0 + 109

Reorder the terms:
-109 + 109 + 1x + x2 = 0 + 109

Combine like terms: -109 + 109 = 0
0 + 1x + x2 = 0 + 109
1x + x2 = 0 + 109

Combine like terms: 0 + 109 = 109
1x + x2 = 109

The x term is 1x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 109 + 0.25

Reorder the terms:
0.25 + x + x2 = 109 + 0.25

Combine like terms: 109 + 0.25 = 109.25
0.25 + x + x2 = 109.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 109.25

Calculate the square root of the right side: 10.45227248

Break this problem into two subproblems by setting 
(x + 0.5) equal to 10.45227248 and -10.45227248.

Subproblem 1
x + 0.5 = 10.45227248

Simplifying
x + 0.5 = 10.45227248

Reorder the terms:
0.5 + x = 10.45227248

Solving
0.5 + x = 10.45227248

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 10.45227248 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 10.45227248 + -0.5
x = 10.45227248 + -0.5

Combine like terms: 10.45227248 + -0.5 = 9.95227248
x = 9.95227248

Simplifying
x = 9.95227248

Subproblem 2
x + 0.5 = -10.45227248

Simplifying
x + 0.5 = -10.45227248

Reorder the terms:
0.5 + x = -10.45227248

Solving
0.5 + x = -10.45227248

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -10.45227248 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -10.45227248 + -0.5
x = -10.45227248 + -0.5

Combine like terms: -10.45227248 + -0.5 = -10.95227248
x = -10.95227248

Simplifying
x = -10.95227248

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.95227248, -10.95227248}