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Simplifying (10 + w)(w) = 1000 Reorder the terms for easier multiplication: w(10 + w) = 1000 (10 * w + w * w) = 1000 (10w + w2) = 1000 Solving 10w + w2 = 1000 Solving for variable 'w'. Reorder the terms: -1000 + 10w + w2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 10w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '1000' to each side of the equation. -1000 + 10w + 1000 + w2 = 0 + 1000 Reorder the terms: -1000 + 1000 + 10w + w2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + 10w + w2 = 0 + 1000 10w + w2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 10w + w2 = 1000 The w term is 10w. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10w + 25 + w2 = 1000 + 25 Reorder the terms: 25 + 10w + w2 = 1000 + 25 Combine like terms: 1000 + 25 = 1025 25 + 10w + w2 = 1025 Factor a perfect square on the left side: (w + 5)(w + 5) = 1025 Calculate the square root of the right side: 32.015621187 Break this problem into two subproblems by setting (w + 5) equal to 32.015621187 and -32.015621187.Subproblem 1
w + 5 = 32.015621187 Simplifying w + 5 = 32.015621187 Reorder the terms: 5 + w = 32.015621187 Solving 5 + w = 32.015621187 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + w = 32.015621187 + -5 Combine like terms: 5 + -5 = 0 0 + w = 32.015621187 + -5 w = 32.015621187 + -5 Combine like terms: 32.015621187 + -5 = 27.015621187 w = 27.015621187 Simplifying w = 27.015621187Subproblem 2
w + 5 = -32.015621187 Simplifying w + 5 = -32.015621187 Reorder the terms: 5 + w = -32.015621187 Solving 5 + w = -32.015621187 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + w = -32.015621187 + -5 Combine like terms: 5 + -5 = 0 0 + w = -32.015621187 + -5 w = -32.015621187 + -5 Combine like terms: -32.015621187 + -5 = -37.015621187 w = -37.015621187 Simplifying w = -37.015621187Solution
The solution to the problem is based on the solutions from the subproblems. w = {27.015621187, -37.015621187}
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