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Simplifying 6000 = w2 + (w + 40) Reorder the terms: 6000 = w2 + (40 + w) Remove parenthesis around (40 + w) 6000 = w2 + 40 + w Reorder the terms: 6000 = 40 + w + w2 Solving 6000 = 40 + w + w2 Solving for variable 'w'. Combine like terms: 6000 + -40 = 5960 5960 + -1w + -1w2 = 40 + w + w2 + -40 + -1w + -1w2 Reorder the terms: 5960 + -1w + -1w2 = 40 + -40 + w + -1w + w2 + -1w2 Combine like terms: 40 + -40 = 0 5960 + -1w + -1w2 = 0 + w + -1w + w2 + -1w2 5960 + -1w + -1w2 = w + -1w + w2 + -1w2 Combine like terms: w + -1w = 0 5960 + -1w + -1w2 = 0 + w2 + -1w2 5960 + -1w + -1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 5960 + -1w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -5960 + w + w2 = 0 Move the constant term to the right: Add '5960' to each side of the equation. -5960 + w + 5960 + w2 = 0 + 5960 Reorder the terms: -5960 + 5960 + w + w2 = 0 + 5960 Combine like terms: -5960 + 5960 = 0 0 + w + w2 = 0 + 5960 w + w2 = 0 + 5960 Combine like terms: 0 + 5960 = 5960 w + w2 = 5960 The w term is w. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + w2 = 5960 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + w2 = 5960 + 0.25 Combine like terms: 5960 + 0.25 = 5960.25 1.25 + w2 = 5960.25 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 5960.25 Calculate the square root of the right side: 77.202655395 Break this problem into two subproblems by setting (w + 0.5) equal to 77.202655395 and -77.202655395.Subproblem 1
w + 0.5 = 77.202655395 Simplifying w + 0.5 = 77.202655395 Reorder the terms: 0.5 + w = 77.202655395 Solving 0.5 + w = 77.202655395 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 77.202655395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 77.202655395 + -0.5 w = 77.202655395 + -0.5 Combine like terms: 77.202655395 + -0.5 = 76.702655395 w = 76.702655395 Simplifying w = 76.702655395Subproblem 2
w + 0.5 = -77.202655395 Simplifying w + 0.5 = -77.202655395 Reorder the terms: 0.5 + w = -77.202655395 Solving 0.5 + w = -77.202655395 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -77.202655395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -77.202655395 + -0.5 w = -77.202655395 + -0.5 Combine like terms: -77.202655395 + -0.5 = -77.702655395 w = -77.702655395 Simplifying w = -77.702655395Solution
The solution to the problem is based on the solutions from the subproblems. w = {76.702655395, -77.702655395}
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