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Simplifying 0.041 = w(1 + -0.59w) 0.041 = (1 * w + -0.59w * w) 0.041 = (1w + -0.59w2) Solving 0.041 = 1w + -0.59w2 Solving for variable 'w'. Reorder the terms: 0.041 + -1w + 0.59w2 = 1w + -1w + -0.59w2 + 0.59w2 Combine like terms: 1w + -1w = 0 0.041 + -1w + 0.59w2 = 0 + -0.59w2 + 0.59w2 0.041 + -1w + 0.59w2 = -0.59w2 + 0.59w2 Combine like terms: -0.59w2 + 0.59w2 = 0.00 0.041 + -1w + 0.59w2 = 0.00 Begin completing the square. Divide all terms by 0.59 the coefficient of the squared term: Divide each side by '0.59'. 0.06949152542 + -1.694915254w + w2 = 0 Move the constant term to the right: Add '-0.06949152542' to each side of the equation. 0.06949152542 + -1.694915254w + -0.06949152542 + w2 = 0 + -0.06949152542 Reorder the terms: 0.06949152542 + -0.06949152542 + -1.694915254w + w2 = 0 + -0.06949152542 Combine like terms: 0.06949152542 + -0.06949152542 = 0.00000000000 0.00000000000 + -1.694915254w + w2 = 0 + -0.06949152542 -1.694915254w + w2 = 0 + -0.06949152542 Combine like terms: 0 + -0.06949152542 = -0.06949152542 -1.694915254w + w2 = -0.06949152542 The w term is -1.694915254w. Take half its coefficient (-0.847457627). Square it (0.7181844296) and add it to both sides. Add '0.7181844296' to each side of the equation. -1.694915254w + 0.7181844296 + w2 = -0.06949152542 + 0.7181844296 Reorder the terms: 0.7181844296 + -1.694915254w + w2 = -0.06949152542 + 0.7181844296 Combine like terms: -0.06949152542 + 0.7181844296 = 0.64869290418 0.7181844296 + -1.694915254w + w2 = 0.64869290418 Factor a perfect square on the left side: (w + -0.847457627)(w + -0.847457627) = 0.64869290418 Calculate the square root of the right side: 0.80541474 Break this problem into two subproblems by setting (w + -0.847457627) equal to 0.80541474 and -0.80541474.Subproblem 1
w + -0.847457627 = 0.80541474 Simplifying w + -0.847457627 = 0.80541474 Reorder the terms: -0.847457627 + w = 0.80541474 Solving -0.847457627 + w = 0.80541474 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.847457627' to each side of the equation. -0.847457627 + 0.847457627 + w = 0.80541474 + 0.847457627 Combine like terms: -0.847457627 + 0.847457627 = 0.000000000 0.000000000 + w = 0.80541474 + 0.847457627 w = 0.80541474 + 0.847457627 Combine like terms: 0.80541474 + 0.847457627 = 1.652872367 w = 1.652872367 Simplifying w = 1.652872367Subproblem 2
w + -0.847457627 = -0.80541474 Simplifying w + -0.847457627 = -0.80541474 Reorder the terms: -0.847457627 + w = -0.80541474 Solving -0.847457627 + w = -0.80541474 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.847457627' to each side of the equation. -0.847457627 + 0.847457627 + w = -0.80541474 + 0.847457627 Combine like terms: -0.847457627 + 0.847457627 = 0.000000000 0.000000000 + w = -0.80541474 + 0.847457627 w = -0.80541474 + 0.847457627 Combine like terms: -0.80541474 + 0.847457627 = 0.042042887 w = 0.042042887 Simplifying w = 0.042042887Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.652872367, 0.042042887}
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