9=-2(w+45)w

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Solution for 9=-2(w+45)w equation: 


Simplifying
9 = -2(w + 45) * w

Reorder the terms:
9 = -2(45 + w) * w

Reorder the terms for easier multiplication:
9 = -2w(45 + w)
9 = (45 * -2w + w * -2w)
9 = (-90w + -2w2)

Solving
9 = -90w + -2w2

Solving for variable 'w'.

Reorder the terms:
9 + 90w + 2w2 = -90w + 90w + -2w2 + 2w2

Combine like terms: -90w + 90w = 0
9 + 90w + 2w2 = 0 + -2w2 + 2w2
9 + 90w + 2w2 = -2w2 + 2w2

Combine like terms: -2w2 + 2w2 = 0
9 + 90w + 2w2 = 0

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

Divide each side by '2'.
4.5 + 45w + w2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 45w + -4.5 + w2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 45w + w2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 45w + w2 = 0 + -4.5
45w + w2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
45w + w2 = -4.5

The w term is 45w.  Take half its coefficient (22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
45w + 506.25 + w2 = -4.5 + 506.25

Reorder the terms:
506.25 + 45w + w2 = -4.5 + 506.25

Combine like terms: -4.5 + 506.25 = 501.75
506.25 + 45w + w2 = 501.75

Factor a perfect square on the left side:
(w + 22.5)(w + 22.5) = 501.75

Calculate the square root of the right side: 22.399776785

Break this problem into two subproblems by setting 
(w + 22.5) equal to 22.399776785 and -22.399776785.


Subproblem 1
w + 22.5 = 22.399776785

Simplifying
w + 22.5 = 22.399776785

Reorder the terms:
22.5 + w = 22.399776785

Solving
22.5 + w = 22.399776785

Solving for variable 'w'.

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

Add '-22.5' to each side of the equation.
22.5 + -22.5 + w = 22.399776785 + -22.5

Combine like terms: 22.5 + -22.5 = 0.0
0.0 + w = 22.399776785 + -22.5
w = 22.399776785 + -22.5

Combine like terms: 22.399776785 + -22.5 = -0.100223215
w = -0.100223215

Simplifying
w = -0.100223215


Subproblem 2
w + 22.5 = -22.399776785

Simplifying
w + 22.5 = -22.399776785

Reorder the terms:
22.5 + w = -22.399776785

Solving
22.5 + w = -22.399776785

Solving for variable 'w'.

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

Add '-22.5' to each side of the equation.
22.5 + -22.5 + w = -22.399776785 + -22.5

Combine like terms: 22.5 + -22.5 = 0.0
0.0 + w = -22.399776785 + -22.5
w = -22.399776785 + -22.5

Combine like terms: -22.399776785 + -22.5 = -44.899776785
w = -44.899776785

Simplifying
w = -44.899776785


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.100223215, -44.899776785}

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