228=w(2w+6)

Solution for 228=w(2w+6) equation:

Simplifying
228 = w(2w + 6)

Reorder the terms:
228 = w(6 + 2w)
228 = (6 * w + 2w * w)
228 = (6w + 2w2)

Solving
228 = 6w + 2w2

Solving for variable 'w'.

Reorder the terms:
228 + -6w + -2w2 = 6w + -6w + 2w2 + -2w2

Combine like terms: 6w + -6w = 0
228 + -6w + -2w2 = 0 + 2w2 + -2w2
228 + -6w + -2w2 = 2w2 + -2w2

Combine like terms: 2w2 + -2w2 = 0
228 + -6w + -2w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(114 + -3w + -1w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(114 + -3w + -1w2)' equal to zero and attempt to solve:

Simplifying
114 + -3w + -1w2 = 0

Solving
114 + -3w + -1w2 = 0

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

Divide each side by '-1'.
-114 + 3w + w2 = 0

Move the constant term to the right:

Add '114' to each side of the equation.
-114 + 3w + 114 + w2 = 0 + 114

Reorder the terms:
-114 + 114 + 3w + w2 = 0 + 114

Combine like terms: -114 + 114 = 0
0 + 3w + w2 = 0 + 114
3w + w2 = 0 + 114

Combine like terms: 0 + 114 = 114
3w + w2 = 114

The w term is 3w.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3w + 2.25 + w2 = 114 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 114 + 2.25

Combine like terms: 114 + 2.25 = 116.25
2.25 + 3w + w2 = 116.25

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 116.25

Calculate the square root of the right side: 10.781929326

Break this problem into two subproblems by setting 
(w + 1.5) equal to 10.781929326 and -10.781929326.

Subproblem 1
w + 1.5 = 10.781929326

Simplifying
w + 1.5 = 10.781929326

Reorder the terms:
1.5 + w = 10.781929326

Solving
1.5 + w = 10.781929326

Solving for variable 'w'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + w = 10.781929326 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = 10.781929326 + -1.5
w = 10.781929326 + -1.5

Combine like terms: 10.781929326 + -1.5 = 9.281929326
w = 9.281929326

Simplifying
w = 9.281929326

Subproblem 2
w + 1.5 = -10.781929326

Simplifying
w + 1.5 = -10.781929326

Reorder the terms:
1.5 + w = -10.781929326

Solving
1.5 + w = -10.781929326

Solving for variable 'w'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + w = -10.781929326 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = -10.781929326 + -1.5
w = -10.781929326 + -1.5

Combine like terms: -10.781929326 + -1.5 = -12.281929326
w = -12.281929326

Simplifying
w = -12.281929326

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {9.281929326, -12.281929326}
Solution
w = {9.281929326, -12.281929326}