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Simplifying w(w + -32) = 10000 Reorder the terms: w(-32 + w) = 10000 (-32 * w + w * w) = 10000 (-32w + w2) = 10000 Solving -32w + w2 = 10000 Solving for variable 'w'. Reorder the terms: -10000 + -32w + w2 = 10000 + -10000 Combine like terms: 10000 + -10000 = 0 -10000 + -32w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '10000' to each side of the equation. -10000 + -32w + 10000 + w2 = 0 + 10000 Reorder the terms: -10000 + 10000 + -32w + w2 = 0 + 10000 Combine like terms: -10000 + 10000 = 0 0 + -32w + w2 = 0 + 10000 -32w + w2 = 0 + 10000 Combine like terms: 0 + 10000 = 10000 -32w + w2 = 10000 The w term is -32w. Take half its coefficient (-16). Square it (256) and add it to both sides. Add '256' to each side of the equation. -32w + 256 + w2 = 10000 + 256 Reorder the terms: 256 + -32w + w2 = 10000 + 256 Combine like terms: 10000 + 256 = 10256 256 + -32w + w2 = 10256 Factor a perfect square on the left side: (w + -16)(w + -16) = 10256 Calculate the square root of the right side: 101.271911209 Break this problem into two subproblems by setting (w + -16) equal to 101.271911209 and -101.271911209.Subproblem 1
w + -16 = 101.271911209 Simplifying w + -16 = 101.271911209 Reorder the terms: -16 + w = 101.271911209 Solving -16 + w = 101.271911209 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + w = 101.271911209 + 16 Combine like terms: -16 + 16 = 0 0 + w = 101.271911209 + 16 w = 101.271911209 + 16 Combine like terms: 101.271911209 + 16 = 117.271911209 w = 117.271911209 Simplifying w = 117.271911209Subproblem 2
w + -16 = -101.271911209 Simplifying w + -16 = -101.271911209 Reorder the terms: -16 + w = -101.271911209 Solving -16 + w = -101.271911209 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '16' to each side of the equation. -16 + 16 + w = -101.271911209 + 16 Combine like terms: -16 + 16 = 0 0 + w = -101.271911209 + 16 w = -101.271911209 + 16 Combine like terms: -101.271911209 + 16 = -85.271911209 w = -85.271911209 Simplifying w = -85.271911209Solution
The solution to the problem is based on the solutions from the subproblems. w = {117.271911209, -85.271911209}
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