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Simplifying 6w2 + 34w + 41 = 0 Reorder the terms: 41 + 34w + 6w2 = 0 Solving 41 + 34w + 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'. 6.833333333 + 5.666666667w + w2 = 0 Move the constant term to the right: Add '-6.833333333' to each side of the equation. 6.833333333 + 5.666666667w + -6.833333333 + w2 = 0 + -6.833333333 Reorder the terms: 6.833333333 + -6.833333333 + 5.666666667w + w2 = 0 + -6.833333333 Combine like terms: 6.833333333 + -6.833333333 = 0.000000000 0.000000000 + 5.666666667w + w2 = 0 + -6.833333333 5.666666667w + w2 = 0 + -6.833333333 Combine like terms: 0 + -6.833333333 = -6.833333333 5.666666667w + w2 = -6.833333333 The w term is 5.666666667w. Take half its coefficient (2.833333334). Square it (8.027777782) and add it to both sides. Add '8.027777782' to each side of the equation. 5.666666667w + 8.027777782 + w2 = -6.833333333 + 8.027777782 Reorder the terms: 8.027777782 + 5.666666667w + w2 = -6.833333333 + 8.027777782 Combine like terms: -6.833333333 + 8.027777782 = 1.194444449 8.027777782 + 5.666666667w + w2 = 1.194444449 Factor a perfect square on the left side: (w + 2.833333334)(w + 2.833333334) = 1.194444449 Calculate the square root of the right side: 1.092906423 Break this problem into two subproblems by setting (w + 2.833333334) equal to 1.092906423 and -1.092906423.Subproblem 1
w + 2.833333334 = 1.092906423 Simplifying w + 2.833333334 = 1.092906423 Reorder the terms: 2.833333334 + w = 1.092906423 Solving 2.833333334 + w = 1.092906423 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + w = 1.092906423 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + w = 1.092906423 + -2.833333334 w = 1.092906423 + -2.833333334 Combine like terms: 1.092906423 + -2.833333334 = -1.740426911 w = -1.740426911 Simplifying w = -1.740426911Subproblem 2
w + 2.833333334 = -1.092906423 Simplifying w + 2.833333334 = -1.092906423 Reorder the terms: 2.833333334 + w = -1.092906423 Solving 2.833333334 + w = -1.092906423 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.833333334' to each side of the equation. 2.833333334 + -2.833333334 + w = -1.092906423 + -2.833333334 Combine like terms: 2.833333334 + -2.833333334 = 0.000000000 0.000000000 + w = -1.092906423 + -2.833333334 w = -1.092906423 + -2.833333334 Combine like terms: -1.092906423 + -2.833333334 = -3.926239757 w = -3.926239757 Simplifying w = -3.926239757Solution
The solution to the problem is based on the solutions from the subproblems. w = {-1.740426911, -3.926239757}
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