(2w+6)w=120

Solution for (2w+6)w=120 equation:

Simplifying
(2w + 6) * w = 120

Reorder the terms:
(6 + 2w) * w = 120

Reorder the terms for easier multiplication:
w(6 + 2w) = 120
(6 * w + 2w * w) = 120
(6w + 2w2) = 120

Solving
6w + 2w2 = 120

Solving for variable 'w'.

Reorder the terms:
-120 + 6w + 2w2 = 120 + -120

Combine like terms: 120 + -120 = 0
-120 + 6w + 2w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-60 + 3w + w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-60 + 3w + w2)' equal to zero and attempt to solve:

Simplifying
-60 + 3w + w2 = 0

Solving
-60 + 3w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + 3w + 60 + w2 = 0 + 60

Reorder the terms:
-60 + 60 + 3w + w2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + 3w + w2 = 0 + 60
3w + w2 = 0 + 60

Combine like terms: 0 + 60 = 60
3w + w2 = 60

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 = 60 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 60 + 2.25

Combine like terms: 60 + 2.25 = 62.25
2.25 + 3w + w2 = 62.25

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 62.25

Calculate the square root of the right side: 7.889866919

Break this problem into two subproblems by setting 
(w + 1.5) equal to 7.889866919 and -7.889866919.

Subproblem 1
w + 1.5 = 7.889866919

Simplifying
w + 1.5 = 7.889866919

Reorder the terms:
1.5 + w = 7.889866919

Solving
1.5 + w = 7.889866919

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 = 7.889866919 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = 7.889866919 + -1.5
w = 7.889866919 + -1.5

Combine like terms: 7.889866919 + -1.5 = 6.389866919
w = 6.389866919

Simplifying
w = 6.389866919

Subproblem 2
w + 1.5 = -7.889866919

Simplifying
w + 1.5 = -7.889866919

Reorder the terms:
1.5 + w = -7.889866919

Solving
1.5 + w = -7.889866919

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 = -7.889866919 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = -7.889866919 + -1.5
w = -7.889866919 + -1.5

Combine like terms: -7.889866919 + -1.5 = -9.389866919
w = -9.389866919

Simplifying
w = -9.389866919

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {6.389866919, -9.389866919}
Solution
w = {6.389866919, -9.389866919}