115=w(w+1)

Solution for 115=w(w+1) equation:

Simplifying
115 = w(w + 1)

Reorder the terms:
115 = w(1 + w)
115 = (1 * w + w * w)
115 = (1w + w2)

Solving
115 = 1w + w2

Solving for variable 'w'.

Reorder the terms:
115 + -1w + -1w2 = 1w + -1w + w2 + -1w2

Combine like terms: 1w + -1w = 0
115 + -1w + -1w2 = 0 + w2 + -1w2
115 + -1w + -1w2 = w2 + -1w2

Combine like terms: w2 + -1w2 = 0
115 + -1w + -1w2 = 0

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

Divide each side by '-1'.
-115 + w + w2 = 0

Move the constant term to the right:

Add '115' to each side of the equation.
-115 + w + 115 + w2 = 0 + 115

Reorder the terms:
-115 + 115 + w + w2 = 0 + 115

Combine like terms: -115 + 115 = 0
0 + w + w2 = 0 + 115
w + w2 = 0 + 115

Combine like terms: 0 + 115 = 115
w + w2 = 115

The w term is w.  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.
 + 0.25 + w2 = 115 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + w2 = 115 + 0.25

Combine like terms: 115 + 0.25 = 115.25
1.25 + w2 = 115.25

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 115.25

Calculate the square root of the right side: 10.735455277

Break this problem into two subproblems by setting 
(w + 0.5) equal to 10.735455277 and -10.735455277.

Subproblem 1
w + 0.5 = 10.735455277

Simplifying
w + 0.5 = 10.735455277

Reorder the terms:
0.5 + w = 10.735455277

Solving
0.5 + w = 10.735455277

Solving for variable 'w'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + w = 10.735455277 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = 10.735455277 + -0.5
w = 10.735455277 + -0.5

Combine like terms: 10.735455277 + -0.5 = 10.235455277
w = 10.235455277

Simplifying
w = 10.235455277

Subproblem 2
w + 0.5 = -10.735455277

Simplifying
w + 0.5 = -10.735455277

Reorder the terms:
0.5 + w = -10.735455277

Solving
0.5 + w = -10.735455277

Solving for variable 'w'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + w = -10.735455277 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = -10.735455277 + -0.5
w = -10.735455277 + -0.5

Combine like terms: -10.735455277 + -0.5 = -11.235455277
w = -11.235455277

Simplifying
w = -11.235455277

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {10.235455277, -11.235455277}