3w^2+27w+7=0

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Solution for 3w^2+27w+7=0 equation: 


Simplifying
3w2 + 27w + 7 = 0

Reorder the terms:
7 + 27w + 3w2 = 0

Solving
7 + 27w + 3w2 = 0

Solving for variable 'w'.

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

Divide each side by '3'.
2.333333333 + 9w + w2 = 0

Move the constant term to the right:

Add '-2.333333333' to each side of the equation.
2.333333333 + 9w + -2.333333333 + w2 = 0 + -2.333333333

Reorder the terms:
2.333333333 + -2.333333333 + 9w + w2 = 0 + -2.333333333

Combine like terms: 2.333333333 + -2.333333333 = 0.000000000
0.000000000 + 9w + w2 = 0 + -2.333333333
9w + w2 = 0 + -2.333333333

Combine like terms: 0 + -2.333333333 = -2.333333333
9w + w2 = -2.333333333

The w term is 9w.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9w + 20.25 + w2 = -2.333333333 + 20.25

Reorder the terms:
20.25 + 9w + w2 = -2.333333333 + 20.25

Combine like terms: -2.333333333 + 20.25 = 17.916666667
20.25 + 9w + w2 = 17.916666667

Factor a perfect square on the left side:
(w + 4.5)(w + 4.5) = 17.916666667

Calculate the square root of the right side: 4.232808366

Break this problem into two subproblems by setting 
(w + 4.5) equal to 4.232808366 and -4.232808366.


Subproblem 1
w + 4.5 = 4.232808366

Simplifying
w + 4.5 = 4.232808366

Reorder the terms:
4.5 + w = 4.232808366

Solving
4.5 + w = 4.232808366

Solving for variable 'w'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + w = 4.232808366 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + w = 4.232808366 + -4.5
w = 4.232808366 + -4.5

Combine like terms: 4.232808366 + -4.5 = -0.267191634
w = -0.267191634

Simplifying
w = -0.267191634


Subproblem 2
w + 4.5 = -4.232808366

Simplifying
w + 4.5 = -4.232808366

Reorder the terms:
4.5 + w = -4.232808366

Solving
4.5 + w = -4.232808366

Solving for variable 'w'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + w = -4.232808366 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + w = -4.232808366 + -4.5
w = -4.232808366 + -4.5

Combine like terms: -4.232808366 + -4.5 = -8.732808366
w = -8.732808366

Simplifying
w = -8.732808366


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.267191634, -8.732808366}

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