(-4w^2-14w+8)-[(6w^2-3w+12)-(13w^2+21w-7)]=

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Solution for (-4w^2-14w+8)-[(6w^2-3w+12)-(13w^2+21w-7)]= equation: 


Simplifying
(-4w2 + -14w + 8) + -1[(6w2 + -3w + 12) + -1(13w2 + 21w + -7)] = 0

Reorder the terms:
(8 + -14w + -4w2) + -1[(6w2 + -3w + 12) + -1(13w2 + 21w + -7)] = 0

Remove parenthesis around (8 + -14w + -4w2)
8 + -14w + -4w2 + -1[(6w2 + -3w + 12) + -1(13w2 + 21w + -7)] = 0

Reorder the terms:
8 + -14w + -4w2 + -1[(12 + -3w + 6w2) + -1(13w2 + 21w + -7)] = 0

Remove parenthesis around (12 + -3w + 6w2)
8 + -14w + -4w2 + -1[12 + -3w + 6w2 + -1(13w2 + 21w + -7)] = 0

Reorder the terms:
8 + -14w + -4w2 + -1[12 + -3w + 6w2 + -1(-7 + 21w + 13w2)] = 0
8 + -14w + -4w2 + -1[12 + -3w + 6w2 + (-7 * -1 + 21w * -1 + 13w2 * -1)] = 0
8 + -14w + -4w2 + -1[12 + -3w + 6w2 + (7 + -21w + -13w2)] = 0

Reorder the terms:
8 + -14w + -4w2 + -1[12 + 7 + -3w + -21w + 6w2 + -13w2] = 0

Combine like terms: 12 + 7 = 19
8 + -14w + -4w2 + -1[19 + -3w + -21w + 6w2 + -13w2] = 0

Combine like terms: -3w + -21w = -24w
8 + -14w + -4w2 + -1[19 + -24w + 6w2 + -13w2] = 0

Combine like terms: 6w2 + -13w2 = -7w2
8 + -14w + -4w2 + -1[19 + -24w + -7w2] = 0
8 + -14w + -4w2 + [19 * -1 + -24w * -1 + -7w2 * -1] = 0
8 + -14w + -4w2 + [-19 + 24w + 7w2] = 0

Reorder the terms:
8 + -19 + -14w + 24w + -4w2 + 7w2 = 0

Combine like terms: 8 + -19 = -11
-11 + -14w + 24w + -4w2 + 7w2 = 0

Combine like terms: -14w + 24w = 10w
-11 + 10w + -4w2 + 7w2 = 0

Combine like terms: -4w2 + 7w2 = 3w2
-11 + 10w + 3w2 = 0

Solving
-11 + 10w + 3w2 = 0

Solving for variable 'w'.

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

Divide each side by '3'.
-3.666666667 + 3.333333333w + w2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + 3.333333333w + 3.666666667 + w2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + 3.333333333w + w2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + 3.333333333w + w2 = 0 + 3.666666667
3.333333333w + w2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
3.333333333w + w2 = 3.666666667

The w term is 3.333333333w.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333w + 2.777777779 + w2 = 3.666666667 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333w + w2 = 3.666666667 + 2.777777779

Combine like terms: 3.666666667 + 2.777777779 = 6.444444446
2.777777779 + 3.333333333w + w2 = 6.444444446

Factor a perfect square on the left side:
(w + 1.666666667)(w + 1.666666667) = 6.444444446

Calculate the square root of the right side: 2.538591036

Break this problem into two subproblems by setting 
(w + 1.666666667) equal to 2.538591036 and -2.538591036.


Subproblem 1
w + 1.666666667 = 2.538591036

Simplifying
w + 1.666666667 = 2.538591036

Reorder the terms:
1.666666667 + w = 2.538591036

Solving
1.666666667 + w = 2.538591036

Solving for variable 'w'.

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

Add '-1.666666667' to each side of the equation.
1.666666667 + -1.666666667 + w = 2.538591036 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + w = 2.538591036 + -1.666666667
w = 2.538591036 + -1.666666667

Combine like terms: 2.538591036 + -1.666666667 = 0.871924369
w = 0.871924369

Simplifying
w = 0.871924369


Subproblem 2
w + 1.666666667 = -2.538591036

Simplifying
w + 1.666666667 = -2.538591036

Reorder the terms:
1.666666667 + w = -2.538591036

Solving
1.666666667 + w = -2.538591036

Solving for variable 'w'.

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

Add '-1.666666667' to each side of the equation.
1.666666667 + -1.666666667 + w = -2.538591036 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + w = -2.538591036 + -1.666666667
w = -2.538591036 + -1.666666667

Combine like terms: -2.538591036 + -1.666666667 = -4.205257703
w = -4.205257703

Simplifying
w = -4.205257703


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.871924369, -4.205257703}

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