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Simplifying w2 + 2w = 1 Reorder the terms: 2w + w2 = 1 Solving 2w + w2 = 1 Solving for variable 'w'. Reorder the terms: -1 + 2w + w2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 2w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 2w + 1 + w2 = 0 + 1 Reorder the terms: -1 + 1 + 2w + w2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2w + w2 = 0 + 1 2w + w2 = 0 + 1 Combine like terms: 0 + 1 = 1 2w + w2 = 1 The w term is 2w. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2w + 1 + w2 = 1 + 1 Reorder the terms: 1 + 2w + w2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + 2w + w2 = 2 Factor a perfect square on the left side: (w + 1)(w + 1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (w + 1) equal to 1.414213562 and -1.414213562.Subproblem 1
w + 1 = 1.414213562 Simplifying w + 1 = 1.414213562 Reorder the terms: 1 + w = 1.414213562 Solving 1 + w = 1.414213562 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + w = 1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + w = 1.414213562 + -1 w = 1.414213562 + -1 Combine like terms: 1.414213562 + -1 = 0.414213562 w = 0.414213562 Simplifying w = 0.414213562Subproblem 2
w + 1 = -1.414213562 Simplifying w + 1 = -1.414213562 Reorder the terms: 1 + w = -1.414213562 Solving 1 + w = -1.414213562 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + w = -1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + w = -1.414213562 + -1 w = -1.414213562 + -1 Combine like terms: -1.414213562 + -1 = -2.414213562 w = -2.414213562 Simplifying w = -2.414213562Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.414213562, -2.414213562}
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