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Simplifying (w + 0.5)(w + -0.4) = 0 Reorder the terms: (0.5 + w)(w + -0.4) = 0 Reorder the terms: (0.5 + w)(-0.4 + w) = 0 Multiply (0.5 + w) * (-0.4 + w) (0.5(-0.4 + w) + w(-0.4 + w)) = 0 ((-0.4 * 0.5 + w * 0.5) + w(-0.4 + w)) = 0 ((-0.2 + 0.5w) + w(-0.4 + w)) = 0 (-0.2 + 0.5w + (-0.4 * w + w * w)) = 0 (-0.2 + 0.5w + (-0.4w + w2)) = 0 Combine like terms: 0.5w + -0.4w = 0.1w (-0.2 + 0.1w + w2) = 0 Solving -0.2 + 0.1w + w2 = 0 Solving for variable 'w'. Begin completing the square. Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 0.1w + 0.2 + w2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + 0.1w + w2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 0.1w + w2 = 0 + 0.2 0.1w + w2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 0.1w + w2 = 0.2 The w term is 0.1w. Take half its coefficient (0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. 0.1w + 0.0025 + w2 = 0.2 + 0.0025 Reorder the terms: 0.0025 + 0.1w + w2 = 0.2 + 0.0025 Combine like terms: 0.2 + 0.0025 = 0.2025 0.0025 + 0.1w + w2 = 0.2025 Factor a perfect square on the left side: (w + 0.05)(w + 0.05) = 0.2025 Calculate the square root of the right side: 0.45 Break this problem into two subproblems by setting (w + 0.05) equal to 0.45 and -0.45.Subproblem 1
w + 0.05 = 0.45 Simplifying w + 0.05 = 0.45 Reorder the terms: 0.05 + w = 0.45 Solving 0.05 + w = 0.45 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.05' to each side of the equation. 0.05 + -0.05 + w = 0.45 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + w = 0.45 + -0.05 w = 0.45 + -0.05 Combine like terms: 0.45 + -0.05 = 0.4 w = 0.4 Simplifying w = 0.4Subproblem 2
w + 0.05 = -0.45 Simplifying w + 0.05 = -0.45 Reorder the terms: 0.05 + w = -0.45 Solving 0.05 + w = -0.45 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.05' to each side of the equation. 0.05 + -0.05 + w = -0.45 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + w = -0.45 + -0.05 w = -0.45 + -0.05 Combine like terms: -0.45 + -0.05 = -0.5 w = -0.5 Simplifying w = -0.5Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.4, -0.5}
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