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Simplifying 241 = 2(w + 16) * w Reorder the terms: 241 = 2(16 + w) * w Reorder the terms for easier multiplication: 241 = 2w(16 + w) 241 = (16 * 2w + w * 2w) 241 = (32w + 2w2) Solving 241 = 32w + 2w2 Solving for variable 'w'. Reorder the terms: 241 + -32w + -2w2 = 32w + -32w + 2w2 + -2w2 Combine like terms: 32w + -32w = 0 241 + -32w + -2w2 = 0 + 2w2 + -2w2 241 + -32w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 241 + -32w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -120.5 + 16w + w2 = 0 Move the constant term to the right: Add '120.5' to each side of the equation. -120.5 + 16w + 120.5 + w2 = 0 + 120.5 Reorder the terms: -120.5 + 120.5 + 16w + w2 = 0 + 120.5 Combine like terms: -120.5 + 120.5 = 0.0 0.0 + 16w + w2 = 0 + 120.5 16w + w2 = 0 + 120.5 Combine like terms: 0 + 120.5 = 120.5 16w + w2 = 120.5 The w term is 16w. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16w + 64 + w2 = 120.5 + 64 Reorder the terms: 64 + 16w + w2 = 120.5 + 64 Combine like terms: 120.5 + 64 = 184.5 64 + 16w + w2 = 184.5 Factor a perfect square on the left side: (w + 8)(w + 8) = 184.5 Calculate the square root of the right side: 13.583077707 Break this problem into two subproblems by setting (w + 8) equal to 13.583077707 and -13.583077707.Subproblem 1
w + 8 = 13.583077707 Simplifying w + 8 = 13.583077707 Reorder the terms: 8 + w = 13.583077707 Solving 8 + w = 13.583077707 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + w = 13.583077707 + -8 Combine like terms: 8 + -8 = 0 0 + w = 13.583077707 + -8 w = 13.583077707 + -8 Combine like terms: 13.583077707 + -8 = 5.583077707 w = 5.583077707 Simplifying w = 5.583077707Subproblem 2
w + 8 = -13.583077707 Simplifying w + 8 = -13.583077707 Reorder the terms: 8 + w = -13.583077707 Solving 8 + w = -13.583077707 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + w = -13.583077707 + -8 Combine like terms: 8 + -8 = 0 0 + w = -13.583077707 + -8 w = -13.583077707 + -8 Combine like terms: -13.583077707 + -8 = -21.583077707 w = -21.583077707 Simplifying w = -21.583077707Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.583077707, -21.583077707}
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