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Simplifying w2 + 45w = 1000 Reorder the terms: 45w + w2 = 1000 Solving 45w + w2 = 1000 Solving for variable 'w'. Reorder the terms: -1000 + 45w + w2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 45w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '1000' to each side of the equation. -1000 + 45w + 1000 + w2 = 0 + 1000 Reorder the terms: -1000 + 1000 + 45w + w2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + 45w + w2 = 0 + 1000 45w + w2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 45w + w2 = 1000 The w term is 45w. Take half its coefficient (22.5). Square it (506.25) and add it to both sides. Add '506.25' to each side of the equation. 45w + 506.25 + w2 = 1000 + 506.25 Reorder the terms: 506.25 + 45w + w2 = 1000 + 506.25 Combine like terms: 1000 + 506.25 = 1506.25 506.25 + 45w + w2 = 1506.25 Factor a perfect square on the left side: (w + 22.5)(w + 22.5) = 1506.25 Calculate the square root of the right side: 38.810436741 Break this problem into two subproblems by setting (w + 22.5) equal to 38.810436741 and -38.810436741.Subproblem 1
w + 22.5 = 38.810436741 Simplifying w + 22.5 = 38.810436741 Reorder the terms: 22.5 + w = 38.810436741 Solving 22.5 + w = 38.810436741 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + w = 38.810436741 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + w = 38.810436741 + -22.5 w = 38.810436741 + -22.5 Combine like terms: 38.810436741 + -22.5 = 16.310436741 w = 16.310436741 Simplifying w = 16.310436741Subproblem 2
w + 22.5 = -38.810436741 Simplifying w + 22.5 = -38.810436741 Reorder the terms: 22.5 + w = -38.810436741 Solving 22.5 + w = -38.810436741 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + w = -38.810436741 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + w = -38.810436741 + -22.5 w = -38.810436741 + -22.5 Combine like terms: -38.810436741 + -22.5 = -61.310436741 w = -61.310436741 Simplifying w = -61.310436741Solution
The solution to the problem is based on the solutions from the subproblems. w = {16.310436741, -61.310436741}
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