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