6w^2+11w-109=0

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Solution for 6w^2+11w-109=0 equation:


Simplifying
6w2 + 11w + -109 = 0

Reorder the terms:
-109 + 11w + 6w2 = 0

Solving
-109 + 11w + 6w2 = 0

Solving for variable 'w'.

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

Divide each side by '6'.
-18.16666667 + 1.833333333w + w2 = 0

Move the constant term to the right:

Add '18.16666667' to each side of the equation.
-18.16666667 + 1.833333333w + 18.16666667 + w2 = 0 + 18.16666667

Reorder the terms:
-18.16666667 + 18.16666667 + 1.833333333w + w2 = 0 + 18.16666667

Combine like terms: -18.16666667 + 18.16666667 = 0.00000000
0.00000000 + 1.833333333w + w2 = 0 + 18.16666667
1.833333333w + w2 = 0 + 18.16666667

Combine like terms: 0 + 18.16666667 = 18.16666667
1.833333333w + w2 = 18.16666667

The w term is 1.833333333w.  Take half its coefficient (0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
1.833333333w + 0.8402777775 + w2 = 18.16666667 + 0.8402777775

Reorder the terms:
0.8402777775 + 1.833333333w + w2 = 18.16666667 + 0.8402777775

Combine like terms: 18.16666667 + 0.8402777775 = 19.0069444475
0.8402777775 + 1.833333333w + w2 = 19.0069444475

Factor a perfect square on the left side:
(w + 0.9166666665)(w + 0.9166666665) = 19.0069444475

Calculate the square root of the right side: 4.359695454

Break this problem into two subproblems by setting 
(w + 0.9166666665) equal to 4.359695454 and -4.359695454.

Subproblem 1

w + 0.9166666665 = 4.359695454 Simplifying w + 0.9166666665 = 4.359695454 Reorder the terms: 0.9166666665 + w = 4.359695454 Solving 0.9166666665 + w = 4.359695454 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + w = 4.359695454 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + w = 4.359695454 + -0.9166666665 w = 4.359695454 + -0.9166666665 Combine like terms: 4.359695454 + -0.9166666665 = 3.4430287875 w = 3.4430287875 Simplifying w = 3.4430287875

Subproblem 2

w + 0.9166666665 = -4.359695454 Simplifying w + 0.9166666665 = -4.359695454 Reorder the terms: 0.9166666665 + w = -4.359695454 Solving 0.9166666665 + w = -4.359695454 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + w = -4.359695454 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + w = -4.359695454 + -0.9166666665 w = -4.359695454 + -0.9166666665 Combine like terms: -4.359695454 + -0.9166666665 = -5.2763621205 w = -5.2763621205 Simplifying w = -5.2763621205

Solution

The solution to the problem is based on the solutions from the subproblems. w = {3.4430287875, -5.2763621205}

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