-36-2w(w+77)=-4(2w+52)

Solution for -36-2w(w+77)=-4(2w+52) equation:

Simplifying
-36 + -2w(w + 77) = -4(2w + 52)

Reorder the terms:
-36 + -2w(77 + w) = -4(2w + 52)
-36 + (77 * -2w + w * -2w) = -4(2w + 52)
-36 + (-154w + -2w2) = -4(2w + 52)

Reorder the terms:
-36 + -154w + -2w2 = -4(52 + 2w)
-36 + -154w + -2w2 = (52 * -4 + 2w * -4)
-36 + -154w + -2w2 = (-208 + -8w)

Solving
-36 + -154w + -2w2 = -208 + -8w

Solving for variable 'w'.

Reorder the terms:
-36 + 208 + -154w + 8w + -2w2 = -208 + -8w + 208 + 8w

Combine like terms: -36 + 208 = 172
172 + -154w + 8w + -2w2 = -208 + -8w + 208 + 8w

Combine like terms: -154w + 8w = -146w
172 + -146w + -2w2 = -208 + -8w + 208 + 8w

Reorder the terms:
172 + -146w + -2w2 = -208 + 208 + -8w + 8w

Combine like terms: -208 + 208 = 0
172 + -146w + -2w2 = 0 + -8w + 8w
172 + -146w + -2w2 = -8w + 8w

Combine like terms: -8w + 8w = 0
172 + -146w + -2w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(86 + -73w + -1w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(86 + -73w + -1w2)' equal to zero and attempt to solve:

Simplifying
86 + -73w + -1w2 = 0

Solving
86 + -73w + -1w2 = 0

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

Divide each side by '-1'.
-86 + 73w + w2 = 0

Move the constant term to the right:

Add '86' to each side of the equation.
-86 + 73w + 86 + w2 = 0 + 86

Reorder the terms:
-86 + 86 + 73w + w2 = 0 + 86

Combine like terms: -86 + 86 = 0
0 + 73w + w2 = 0 + 86
73w + w2 = 0 + 86

Combine like terms: 0 + 86 = 86
73w + w2 = 86

The w term is 73w.  Take half its coefficient (36.5).
Square it (1332.25) and add it to both sides.

Add '1332.25' to each side of the equation.
73w + 1332.25 + w2 = 86 + 1332.25

Reorder the terms:
1332.25 + 73w + w2 = 86 + 1332.25

Combine like terms: 86 + 1332.25 = 1418.25
1332.25 + 73w + w2 = 1418.25

Factor a perfect square on the left side:
(w + 36.5)(w + 36.5) = 1418.25

Calculate the square root of the right side: 37.659660115

Break this problem into two subproblems by setting 
(w + 36.5) equal to 37.659660115 and -37.659660115.

Subproblem 1
w + 36.5 = 37.659660115

Simplifying
w + 36.5 = 37.659660115

Reorder the terms:
36.5 + w = 37.659660115

Solving
36.5 + w = 37.659660115

Solving for variable 'w'.

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

Add '-36.5' to each side of the equation.
36.5 + -36.5 + w = 37.659660115 + -36.5

Combine like terms: 36.5 + -36.5 = 0.0
0.0 + w = 37.659660115 + -36.5
w = 37.659660115 + -36.5

Combine like terms: 37.659660115 + -36.5 = 1.159660115
w = 1.159660115

Simplifying
w = 1.159660115

Subproblem 2
w + 36.5 = -37.659660115

Simplifying
w + 36.5 = -37.659660115

Reorder the terms:
36.5 + w = -37.659660115

Solving
36.5 + w = -37.659660115

Solving for variable 'w'.

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

Add '-36.5' to each side of the equation.
36.5 + -36.5 + w = -37.659660115 + -36.5

Combine like terms: 36.5 + -36.5 = 0.0
0.0 + w = -37.659660115 + -36.5
w = -37.659660115 + -36.5

Combine like terms: -37.659660115 + -36.5 = -74.159660115
w = -74.159660115

Simplifying
w = -74.159660115

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {1.159660115, -74.159660115}
Solution
w = {1.159660115, -74.159660115}