(w)=12w^2+17w-40

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Solution for (w)=12w^2+17w-40 equation: 


Simplifying
(w) = 12w2 + 17w + -40
w = 12w2 + 17w + -40

Reorder the terms:
w = -40 + 17w + 12w2

Solving
w = -40 + 17w + 12w2

Solving for variable 'w'.

Reorder the terms:
40 + w + -17w + -12w2 = -40 + 17w + 12w2 + 40 + -17w + -12w2

Combine like terms: w + -17w = -16w
40 + -16w + -12w2 = -40 + 17w + 12w2 + 40 + -17w + -12w2

Reorder the terms:
40 + -16w + -12w2 = -40 + 40 + 17w + -17w + 12w2 + -12w2

Combine like terms: -40 + 40 = 0
40 + -16w + -12w2 = 0 + 17w + -17w + 12w2 + -12w2
40 + -16w + -12w2 = 17w + -17w + 12w2 + -12w2

Combine like terms: 17w + -17w = 0
40 + -16w + -12w2 = 0 + 12w2 + -12w2
40 + -16w + -12w2 = 12w2 + -12w2

Combine like terms: 12w2 + -12w2 = 0
40 + -16w + -12w2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(10 + -4w + -3w2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(10 + -4w + -3w2)' equal to zero and attempt to solve:

Simplifying
10 + -4w + -3w2 = 0

Solving
10 + -4w + -3w2 = 0

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

Divide each side by '-3'.
-3.333333333 + 1.333333333w + w2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + 1.333333333w + 3.333333333 + w2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + 1.333333333w + w2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + 1.333333333w + w2 = 0 + 3.333333333
1.333333333w + w2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
1.333333333w + w2 = 3.333333333

The w term is 1.333333333w.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333w + 0.4444444442 + w2 = 3.333333333 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333w + w2 = 3.333333333 + 0.4444444442

Combine like terms: 3.333333333 + 0.4444444442 = 3.7777777772
0.4444444442 + 1.333333333w + w2 = 3.7777777772

Factor a perfect square on the left side:
(w + 0.6666666665)(w + 0.6666666665) = 3.7777777772

Calculate the square root of the right side: 1.943650631

Break this problem into two subproblems by setting 
(w + 0.6666666665) equal to 1.943650631 and -1.943650631.


Subproblem 1
w + 0.6666666665 = 1.943650631

Simplifying
w + 0.6666666665 = 1.943650631

Reorder the terms:
0.6666666665 + w = 1.943650631

Solving
0.6666666665 + w = 1.943650631

Solving for variable 'w'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + w = 1.943650631 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + w = 1.943650631 + -0.6666666665
w = 1.943650631 + -0.6666666665

Combine like terms: 1.943650631 + -0.6666666665 = 1.2769839645
w = 1.2769839645

Simplifying
w = 1.2769839645


Subproblem 2
w + 0.6666666665 = -1.943650631

Simplifying
w + 0.6666666665 = -1.943650631

Reorder the terms:
0.6666666665 + w = -1.943650631

Solving
0.6666666665 + w = -1.943650631

Solving for variable 'w'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + w = -1.943650631 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + w = -1.943650631 + -0.6666666665
w = -1.943650631 + -0.6666666665

Combine like terms: -1.943650631 + -0.6666666665 = -2.6103172975
w = -2.6103172975

Simplifying
w = -2.6103172975


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {1.2769839645, -2.6103172975}
Solution
w = {1.2769839645, -2.6103172975}

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