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Simplifying (w + 3) * w = 104 Reorder the terms: (3 + w) * w = 104 Reorder the terms for easier multiplication: w(3 + w) = 104 (3 * w + w * w) = 104 (3w + w2) = 104 Solving 3w + w2 = 104 Solving for variable 'w'. Reorder the terms: -104 + 3w + w2 = 104 + -104 Combine like terms: 104 + -104 = 0 -104 + 3w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '104' to each side of the equation. -104 + 3w + 104 + w2 = 0 + 104 Reorder the terms: -104 + 104 + 3w + w2 = 0 + 104 Combine like terms: -104 + 104 = 0 0 + 3w + w2 = 0 + 104 3w + w2 = 0 + 104 Combine like terms: 0 + 104 = 104 3w + w2 = 104 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 104 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 104 + 2.25 Combine like terms: 104 + 2.25 = 106.25 2.25 + 3w + w2 = 106.25 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 106.25 Calculate the square root of the right side: 10.307764064 Break this problem into two subproblems by setting (w + 1.5) equal to 10.307764064 and -10.307764064.Subproblem 1
w + 1.5 = 10.307764064 Simplifying w + 1.5 = 10.307764064 Reorder the terms: 1.5 + w = 10.307764064 Solving 1.5 + w = 10.307764064 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 10.307764064 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 10.307764064 + -1.5 w = 10.307764064 + -1.5 Combine like terms: 10.307764064 + -1.5 = 8.807764064 w = 8.807764064 Simplifying w = 8.807764064Subproblem 2
w + 1.5 = -10.307764064 Simplifying w + 1.5 = -10.307764064 Reorder the terms: 1.5 + w = -10.307764064 Solving 1.5 + w = -10.307764064 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -10.307764064 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -10.307764064 + -1.5 w = -10.307764064 + -1.5 Combine like terms: -10.307764064 + -1.5 = -11.807764064 w = -11.807764064 Simplifying w = -11.807764064Solution
The solution to the problem is based on the solutions from the subproblems. w = {8.807764064, -11.807764064}
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