-3w^2+8w+22=0

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Solution for -3w^2+8w+22=0 equation: 


Simplifying
-3w2 + 8w + 22 = 0

Reorder the terms:
22 + 8w + -3w2 = 0

Solving
22 + 8w + -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'.
-7.333333333 + -2.666666667w + w2 = 0

Move the constant term to the right:

Add '7.333333333' to each side of the equation.
-7.333333333 + -2.666666667w + 7.333333333 + w2 = 0 + 7.333333333

Reorder the terms:
-7.333333333 + 7.333333333 + -2.666666667w + w2 = 0 + 7.333333333

Combine like terms: -7.333333333 + 7.333333333 = 0.000000000
0.000000000 + -2.666666667w + w2 = 0 + 7.333333333
-2.666666667w + w2 = 0 + 7.333333333

Combine like terms: 0 + 7.333333333 = 7.333333333
-2.666666667w + w2 = 7.333333333

The w term is -2.666666667w.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667w + 1.777777780 + w2 = 7.333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667w + w2 = 7.333333333 + 1.777777780

Combine like terms: 7.333333333 + 1.777777780 = 9.111111113
1.777777780 + -2.666666667w + w2 = 9.111111113

Factor a perfect square on the left side:
(w + -1.333333334)(w + -1.333333334) = 9.111111113

Calculate the square root of the right side: 3.018461713

Break this problem into two subproblems by setting 
(w + -1.333333334) equal to 3.018461713 and -3.018461713.


Subproblem 1
w + -1.333333334 = 3.018461713

Simplifying
w + -1.333333334 = 3.018461713

Reorder the terms:
-1.333333334 + w = 3.018461713

Solving
-1.333333334 + w = 3.018461713

Solving for variable 'w'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + w = 3.018461713 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + w = 3.018461713 + 1.333333334
w = 3.018461713 + 1.333333334

Combine like terms: 3.018461713 + 1.333333334 = 4.351795047
w = 4.351795047

Simplifying
w = 4.351795047


Subproblem 2
w + -1.333333334 = -3.018461713

Simplifying
w + -1.333333334 = -3.018461713

Reorder the terms:
-1.333333334 + w = -3.018461713

Solving
-1.333333334 + w = -3.018461713

Solving for variable 'w'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + w = -3.018461713 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + w = -3.018461713 + 1.333333334
w = -3.018461713 + 1.333333334

Combine like terms: -3.018461713 + 1.333333334 = -1.685128379
w = -1.685128379

Simplifying
w = -1.685128379


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {4.351795047, -1.685128379}

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