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Simplifying (5 + w) * w = 102 Reorder the terms for easier multiplication: w(5 + w) = 102 (5 * w + w * w) = 102 (5w + w2) = 102 Solving 5w + w2 = 102 Solving for variable 'w'. Reorder the terms: -102 + 5w + w2 = 102 + -102 Combine like terms: 102 + -102 = 0 -102 + 5w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '102' to each side of the equation. -102 + 5w + 102 + w2 = 0 + 102 Reorder the terms: -102 + 102 + 5w + w2 = 0 + 102 Combine like terms: -102 + 102 = 0 0 + 5w + w2 = 0 + 102 5w + w2 = 0 + 102 Combine like terms: 0 + 102 = 102 5w + w2 = 102 The w term is 5w. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5w + 6.25 + w2 = 102 + 6.25 Reorder the terms: 6.25 + 5w + w2 = 102 + 6.25 Combine like terms: 102 + 6.25 = 108.25 6.25 + 5w + w2 = 108.25 Factor a perfect square on the left side: (w + 2.5)(w + 2.5) = 108.25 Calculate the square root of the right side: 10.404326023 Break this problem into two subproblems by setting (w + 2.5) equal to 10.404326023 and -10.404326023.Subproblem 1
w + 2.5 = 10.404326023 Simplifying w + 2.5 = 10.404326023 Reorder the terms: 2.5 + w = 10.404326023 Solving 2.5 + w = 10.404326023 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = 10.404326023 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = 10.404326023 + -2.5 w = 10.404326023 + -2.5 Combine like terms: 10.404326023 + -2.5 = 7.904326023 w = 7.904326023 Simplifying w = 7.904326023Subproblem 2
w + 2.5 = -10.404326023 Simplifying w + 2.5 = -10.404326023 Reorder the terms: 2.5 + w = -10.404326023 Solving 2.5 + w = -10.404326023 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = -10.404326023 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = -10.404326023 + -2.5 w = -10.404326023 + -2.5 Combine like terms: -10.404326023 + -2.5 = -12.904326023 w = -12.904326023 Simplifying w = -12.904326023Solution
The solution to the problem is based on the solutions from the subproblems. w = {7.904326023, -12.904326023}
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