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Simplifying (2w + 1)(w) = 88 Reorder the terms: (1 + 2w)(w) = 88 Reorder the terms for easier multiplication: w(1 + 2w) = 88 (1 * w + 2w * w) = 88 (1w + 2w2) = 88 Solving 1w + 2w2 = 88 Solving for variable 'w'. Reorder the terms: -88 + 1w + 2w2 = 88 + -88 Combine like terms: 88 + -88 = 0 -88 + 1w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -44 + 0.5w + w2 = 0 Move the constant term to the right: Add '44' to each side of the equation. -44 + 0.5w + 44 + w2 = 0 + 44 Reorder the terms: -44 + 44 + 0.5w + w2 = 0 + 44 Combine like terms: -44 + 44 = 0 0 + 0.5w + w2 = 0 + 44 0.5w + w2 = 0 + 44 Combine like terms: 0 + 44 = 44 0.5w + w2 = 44 The w term is 0.5w. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5w + 0.0625 + w2 = 44 + 0.0625 Reorder the terms: 0.0625 + 0.5w + w2 = 44 + 0.0625 Combine like terms: 44 + 0.0625 = 44.0625 0.0625 + 0.5w + w2 = 44.0625 Factor a perfect square on the left side: (w + 0.25)(w + 0.25) = 44.0625 Calculate the square root of the right side: 6.637959024 Break this problem into two subproblems by setting (w + 0.25) equal to 6.637959024 and -6.637959024.Subproblem 1
w + 0.25 = 6.637959024 Simplifying w + 0.25 = 6.637959024 Reorder the terms: 0.25 + w = 6.637959024 Solving 0.25 + w = 6.637959024 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = 6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = 6.637959024 + -0.25 w = 6.637959024 + -0.25 Combine like terms: 6.637959024 + -0.25 = 6.387959024 w = 6.387959024 Simplifying w = 6.387959024Subproblem 2
w + 0.25 = -6.637959024 Simplifying w + 0.25 = -6.637959024 Reorder the terms: 0.25 + w = -6.637959024 Solving 0.25 + w = -6.637959024 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = -6.637959024 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = -6.637959024 + -0.25 w = -6.637959024 + -0.25 Combine like terms: -6.637959024 + -0.25 = -6.887959024 w = -6.887959024 Simplifying w = -6.887959024Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.387959024, -6.887959024}
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