w(10w+120)+(45w+25)=

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Solution for w(10w+120)+(45w+25)= equation:


Simplifying
w(10w + 120) + (45w + 25) = 0

Reorder the terms:
w(120 + 10w) + (45w + 25) = 0
(120 * w + 10w * w) + (45w + 25) = 0
(120w + 10w2) + (45w + 25) = 0

Reorder the terms:
120w + 10w2 + (25 + 45w) = 0

Remove parenthesis around (25 + 45w)
120w + 10w2 + 25 + 45w = 0

Reorder the terms:
25 + 120w + 45w + 10w2 = 0

Combine like terms: 120w + 45w = 165w
25 + 165w + 10w2 = 0

Solving
25 + 165w + 10w2 = 0

Solving for variable 'w'.

Factor out the Greatest Common Factor (GCF), '5'.
5(5 + 33w + 2w2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(5 + 33w + 2w2)' equal to zero and attempt to solve:

Simplifying
5 + 33w + 2w2 = 0

Solving
5 + 33w + 2w2 = 0

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

Divide each side by '2'.
2.5 + 16.5w + w2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 16.5w + -2.5 + w2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 16.5w + w2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 16.5w + w2 = 0 + -2.5
16.5w + w2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
16.5w + w2 = -2.5

The w term is 16.5w.  Take half its coefficient (8.25).
Square it (68.0625) and add it to both sides.

Add '68.0625' to each side of the equation.
16.5w + 68.0625 + w2 = -2.5 + 68.0625

Reorder the terms:
68.0625 + 16.5w + w2 = -2.5 + 68.0625

Combine like terms: -2.5 + 68.0625 = 65.5625
68.0625 + 16.5w + w2 = 65.5625

Factor a perfect square on the left side:
(w + 8.25)(w + 8.25) = 65.5625

Calculate the square root of the right side: 8.09706737

Break this problem into two subproblems by setting 
(w + 8.25) equal to 8.09706737 and -8.09706737.


Subproblem 1
w + 8.25 = 8.09706737

Simplifying
w + 8.25 = 8.09706737

Reorder the terms:
8.25 + w = 8.09706737

Solving
8.25 + w = 8.09706737

Solving for variable 'w'.

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

Add '-8.25' to each side of the equation.
8.25 + -8.25 + w = 8.09706737 + -8.25

Combine like terms: 8.25 + -8.25 = 0.00
0.00 + w = 8.09706737 + -8.25
w = 8.09706737 + -8.25

Combine like terms: 8.09706737 + -8.25 = -0.15293263
w = -0.15293263

Simplifying
w = -0.15293263


Subproblem 2
w + 8.25 = -8.09706737

Simplifying
w + 8.25 = -8.09706737

Reorder the terms:
8.25 + w = -8.09706737

Solving
8.25 + w = -8.09706737

Solving for variable 'w'.

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

Add '-8.25' to each side of the equation.
8.25 + -8.25 + w = -8.09706737 + -8.25

Combine like terms: 8.25 + -8.25 = 0.00
0.00 + w = -8.09706737 + -8.25
w = -8.09706737 + -8.25

Combine like terms: -8.09706737 + -8.25 = -16.34706737
w = -16.34706737

Simplifying
w = -16.34706737


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.15293263, -16.34706737}
Solution
w = {-0.15293263, -16.34706737}

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