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Simplifying -1 + 9(w2 + -9w + -5) + -6w2 = 0 Reorder the terms: -1 + 9(-5 + -9w + w2) + -6w2 = 0 -1 + (-5 * 9 + -9w * 9 + w2 * 9) + -6w2 = 0 -1 + (-45 + -81w + 9w2) + -6w2 = 0 Combine like terms: -1 + -45 = -46 -46 + -81w + 9w2 + -6w2 = 0 Combine like terms: 9w2 + -6w2 = 3w2 -46 + -81w + 3w2 = 0 Solving -46 + -81w + 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'. -15.33333333 + -27w + w2 = 0 Move the constant term to the right: Add '15.33333333' to each side of the equation. -15.33333333 + -27w + 15.33333333 + w2 = 0 + 15.33333333 Reorder the terms: -15.33333333 + 15.33333333 + -27w + w2 = 0 + 15.33333333 Combine like terms: -15.33333333 + 15.33333333 = 0.00000000 0.00000000 + -27w + w2 = 0 + 15.33333333 -27w + w2 = 0 + 15.33333333 Combine like terms: 0 + 15.33333333 = 15.33333333 -27w + w2 = 15.33333333 The w term is -27w. Take half its coefficient (-13.5). Square it (182.25) and add it to both sides. Add '182.25' to each side of the equation. -27w + 182.25 + w2 = 15.33333333 + 182.25 Reorder the terms: 182.25 + -27w + w2 = 15.33333333 + 182.25 Combine like terms: 15.33333333 + 182.25 = 197.58333333 182.25 + -27w + w2 = 197.58333333 Factor a perfect square on the left side: (w + -13.5)(w + -13.5) = 197.58333333 Calculate the square root of the right side: 14.056433877 Break this problem into two subproblems by setting (w + -13.5) equal to 14.056433877 and -14.056433877.Subproblem 1
w + -13.5 = 14.056433877 Simplifying w + -13.5 = 14.056433877 Reorder the terms: -13.5 + w = 14.056433877 Solving -13.5 + w = 14.056433877 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + w = 14.056433877 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + w = 14.056433877 + 13.5 w = 14.056433877 + 13.5 Combine like terms: 14.056433877 + 13.5 = 27.556433877 w = 27.556433877 Simplifying w = 27.556433877Subproblem 2
w + -13.5 = -14.056433877 Simplifying w + -13.5 = -14.056433877 Reorder the terms: -13.5 + w = -14.056433877 Solving -13.5 + w = -14.056433877 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + w = -14.056433877 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + w = -14.056433877 + 13.5 w = -14.056433877 + 13.5 Combine like terms: -14.056433877 + 13.5 = -0.556433877 w = -0.556433877 Simplifying w = -0.556433877Solution
The solution to the problem is based on the solutions from the subproblems. w = {27.556433877, -0.556433877}
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