x(x+1)=105

Solution for x(x+1)=105 equation:

Simplifying
x(x + 1) = 105

Reorder the terms:
x(1 + x) = 105
(1 * x + x * x) = 105
(1x + x2) = 105

Solving
1x + x2 = 105

Solving for variable 'x'.

Reorder the terms:
-105 + 1x + x2 = 105 + -105

Combine like terms: 105 + -105 = 0
-105 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '105' to each side of the equation.
-105 + 1x + 105 + x2 = 0 + 105

Reorder the terms:
-105 + 105 + 1x + x2 = 0 + 105

Combine like terms: -105 + 105 = 0
0 + 1x + x2 = 0 + 105
1x + x2 = 0 + 105

Combine like terms: 0 + 105 = 105
1x + x2 = 105

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 = 105 + 0.25

Reorder the terms:
0.25 + x + x2 = 105 + 0.25

Combine like terms: 105 + 0.25 = 105.25
0.25 + x + x2 = 105.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 105.25

Calculate the square root of the right side: 10.259142264

Break this problem into two subproblems by setting 
(x + 0.5) equal to 10.259142264 and -10.259142264.

Subproblem 1
x + 0.5 = 10.259142264

Simplifying
x + 0.5 = 10.259142264

Reorder the terms:
0.5 + x = 10.259142264

Solving
0.5 + x = 10.259142264

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 = 10.259142264 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 10.259142264 + -0.5
x = 10.259142264 + -0.5

Combine like terms: 10.259142264 + -0.5 = 9.759142264
x = 9.759142264

Simplifying
x = 9.759142264

Subproblem 2
x + 0.5 = -10.259142264

Simplifying
x + 0.5 = -10.259142264

Reorder the terms:
0.5 + x = -10.259142264

Solving
0.5 + x = -10.259142264

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 = -10.259142264 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -10.259142264 + -0.5
x = -10.259142264 + -0.5

Combine like terms: -10.259142264 + -0.5 = -10.759142264
x = -10.759142264

Simplifying
x = -10.759142264

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.759142264, -10.759142264}