(6+2w)w=96

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Solution for (6+2w)w=96 equation: 


Simplifying
(6 + 2w) * w = 96

Reorder the terms for easier multiplication:
w(6 + 2w) = 96
(6 * w + 2w * w) = 96
(6w + 2w2) = 96

Solving
6w + 2w2 = 96

Solving for variable 'w'.

Reorder the terms:
-96 + 6w + 2w2 = 96 + -96

Combine like terms: 96 + -96 = 0
-96 + 6w + 2w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-48 + 3w + w2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-48 + 3w + w2)' equal to zero and attempt to solve:

Simplifying
-48 + 3w + w2 = 0

Solving
-48 + 3w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '48' to each side of the equation.
-48 + 3w + 48 + w2 = 0 + 48

Reorder the terms:
-48 + 48 + 3w + w2 = 0 + 48

Combine like terms: -48 + 48 = 0
0 + 3w + w2 = 0 + 48
3w + w2 = 0 + 48

Combine like terms: 0 + 48 = 48
3w + w2 = 48

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 = 48 + 2.25

Reorder the terms:
2.25 + 3w + w2 = 48 + 2.25

Combine like terms: 48 + 2.25 = 50.25
2.25 + 3w + w2 = 50.25

Factor a perfect square on the left side:
(w + 1.5)(w + 1.5) = 50.25

Calculate the square root of the right side: 7.088723439

Break this problem into two subproblems by setting 
(w + 1.5) equal to 7.088723439 and -7.088723439.


Subproblem 1
w + 1.5 = 7.088723439

Simplifying
w + 1.5 = 7.088723439

Reorder the terms:
1.5 + w = 7.088723439

Solving
1.5 + w = 7.088723439

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 = 7.088723439 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = 7.088723439 + -1.5
w = 7.088723439 + -1.5

Combine like terms: 7.088723439 + -1.5 = 5.588723439
w = 5.588723439

Simplifying
w = 5.588723439


Subproblem 2
w + 1.5 = -7.088723439

Simplifying
w + 1.5 = -7.088723439

Reorder the terms:
1.5 + w = -7.088723439

Solving
1.5 + w = -7.088723439

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 = -7.088723439 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + w = -7.088723439 + -1.5
w = -7.088723439 + -1.5

Combine like terms: -7.088723439 + -1.5 = -8.588723439
w = -8.588723439

Simplifying
w = -8.588723439


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {5.588723439, -8.588723439}
Solution
w = {5.588723439, -8.588723439}

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