w(2w+1)=15*15

Solution for w(2w+1)=15*15 equation:

Simplifying
w(2w + 1) = 15 * 15

Reorder the terms:
w(1 + 2w) = 15 * 15
(1 * w + 2w * w) = 15 * 15
(1w + 2w2) = 15 * 15

Multiply 15 * 15
1w + 2w2 = 225

Solving
1w + 2w2 = 225

Solving for variable 'w'.

Reorder the terms:
-225 + 1w + 2w2 = 225 + -225

Combine like terms: 225 + -225 = 0
-225 + 1w + 2w2 = 0

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

Divide each side by '2'.
-112.5 + 0.5w + w2 = 0

Move the constant term to the right:

Add '112.5' to each side of the equation.
-112.5 + 0.5w + 112.5 + w2 = 0 + 112.5

Reorder the terms:
-112.5 + 112.5 + 0.5w + w2 = 0 + 112.5

Combine like terms: -112.5 + 112.5 = 0.0
0.0 + 0.5w + w2 = 0 + 112.5
0.5w + w2 = 0 + 112.5

Combine like terms: 0 + 112.5 = 112.5
0.5w + w2 = 112.5

The w term is 0.5w.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5w + 0.0625 + w2 = 112.5 + 0.0625

Reorder the terms:
0.0625 + 0.5w + w2 = 112.5 + 0.0625

Combine like terms: 112.5 + 0.0625 = 112.5625
0.0625 + 0.5w + w2 = 112.5625

Factor a perfect square on the left side:
(w + 0.25)(w + 0.25) = 112.5625

Calculate the square root of the right side: 10.609547587

Break this problem into two subproblems by setting 
(w + 0.25) equal to 10.609547587 and -10.609547587.

Subproblem 1
w + 0.25 = 10.609547587

Simplifying
w + 0.25 = 10.609547587

Reorder the terms:
0.25 + w = 10.609547587

Solving
0.25 + w = 10.609547587

Solving for variable 'w'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + w = 10.609547587 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + w = 10.609547587 + -0.25
w = 10.609547587 + -0.25

Combine like terms: 10.609547587 + -0.25 = 10.359547587
w = 10.359547587

Simplifying
w = 10.359547587

Subproblem 2
w + 0.25 = -10.609547587

Simplifying
w + 0.25 = -10.609547587

Reorder the terms:
0.25 + w = -10.609547587

Solving
0.25 + w = -10.609547587

Solving for variable 'w'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + w = -10.609547587 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + w = -10.609547587 + -0.25
w = -10.609547587 + -0.25

Combine like terms: -10.609547587 + -0.25 = -10.859547587
w = -10.859547587

Simplifying
w = -10.859547587

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {10.359547587, -10.859547587}