(2w+6)w=180

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Solution for (2w+6)w=180 equation: 


Simplifying
(2w + 6) * w = 180

Reorder the terms:
(6 + 2w) * w = 180

Reorder the terms for easier multiplication:
w(6 + 2w) = 180
(6 * w + 2w * w) = 180
(6w + 2w2) = 180

Solving
6w + 2w2 = 180

Solving for variable 'w'.

Reorder the terms:
-180 + 6w + 2w2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 6w + 2w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-90 + 3w + w2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-90 + 3w + w2)' equal to zero and attempt to solve:

Simplifying
-90 + 3w + w2 = 0

Solving
-90 + 3w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + 3w + 90 + w2 = 0 + 90

Reorder the terms:
-90 + 90 + 3w + w2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + 3w + w2 = 0 + 90
3w + w2 = 0 + 90

Combine like terms: 0 + 90 = 90
3w + w2 = 90

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 = 90 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 90 + 2.25

Combine like terms: 90 + 2.25 = 92.25
2.25 + 3w + w2 = 92.25

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 92.25

Calculate the square root of the right side: 9.604686356

Break this problem into two subproblems by setting 
(w + 1.5) equal to 9.604686356 and -9.604686356.


Subproblem 1
w + 1.5 = 9.604686356

Simplifying
w + 1.5 = 9.604686356

Reorder the terms:
1.5 + w = 9.604686356

Solving
1.5 + w = 9.604686356

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 = 9.604686356 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = 9.604686356 + -1.5
w = 9.604686356 + -1.5

Combine like terms: 9.604686356 + -1.5 = 8.104686356
w = 8.104686356

Simplifying
w = 8.104686356


Subproblem 2
w + 1.5 = -9.604686356

Simplifying
w + 1.5 = -9.604686356

Reorder the terms:
1.5 + w = -9.604686356

Solving
1.5 + w = -9.604686356

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 = -9.604686356 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = -9.604686356 + -1.5
w = -9.604686356 + -1.5

Combine like terms: -9.604686356 + -1.5 = -11.104686356
w = -11.104686356

Simplifying
w = -11.104686356


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {8.104686356, -11.104686356}
Solution
w = {8.104686356, -11.104686356}

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