88=(2w+1)w

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Solution for 88=(2w+1)w equation: 


Simplifying
88 = (2w + 1) * w

Reorder the terms:
88 = (1 + 2w) * w

Reorder the terms for easier multiplication:
88 = w(1 + 2w)
88 = (1 * w + 2w * w)
88 = (1w + 2w2)

Solving
88 = 1w + 2w2

Solving for variable 'w'.

Reorder the terms:
88 + -1w + -2w2 = 1w + -1w + 2w2 + -2w2

Combine like terms: 1w + -1w = 0
88 + -1w + -2w2 = 0 + 2w2 + -2w2
88 + -1w + -2w2 = 2w2 + -2w2

Combine like terms: 2w2 + -2w2 = 0
88 + -1w + -2w2 = 0

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

Divide each side by '-2'.
-44 + 0.5w + w2 = 0

Move the constant term to the right:

Add '44' to each side of the equation.
-44 + 0.5w + 44 + w2 = 0 + 44

Reorder the terms:
-44 + 44 + 0.5w + w2 = 0 + 44

Combine like terms: -44 + 44 = 0
0 + 0.5w + w2 = 0 + 44
0.5w + w2 = 0 + 44

Combine like terms: 0 + 44 = 44
0.5w + w2 = 44

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 = 44 + 0.0625

Reorder the terms:
0.0625 + 0.5w + w2 = 44 + 0.0625

Combine like terms: 44 + 0.0625 = 44.0625
0.0625 + 0.5w + w2 = 44.0625

Factor a perfect square on the left side:
(w + 0.25)(w + 0.25) = 44.0625

Calculate the square root of the right side: 6.637959024

Break this problem into two subproblems by setting 
(w + 0.25) equal to 6.637959024 and -6.637959024.


Subproblem 1
w + 0.25 = 6.637959024

Simplifying
w + 0.25 = 6.637959024

Reorder the terms:
0.25 + w = 6.637959024

Solving
0.25 + w = 6.637959024

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 = 6.637959024 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + w = 6.637959024 + -0.25
w = 6.637959024 + -0.25

Combine like terms: 6.637959024 + -0.25 = 6.387959024
w = 6.387959024

Simplifying
w = 6.387959024


Subproblem 2
w + 0.25 = -6.637959024

Simplifying
w + 0.25 = -6.637959024

Reorder the terms:
0.25 + w = -6.637959024

Solving
0.25 + w = -6.637959024

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 = -6.637959024 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + w = -6.637959024 + -0.25
w = -6.637959024 + -0.25

Combine like terms: -6.637959024 + -0.25 = -6.887959024
w = -6.887959024

Simplifying
w = -6.887959024


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {6.387959024, -6.887959024}

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