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Simplifying (w + 0.25)(w + 0.2) = 0 Reorder the terms: (0.25 + w)(w + 0.2) = 0 Reorder the terms: (0.25 + w)(0.2 + w) = 0 Multiply (0.25 + w) * (0.2 + w) (0.25(0.2 + w) + w(0.2 + w)) = 0 ((0.2 * 0.25 + w * 0.25) + w(0.2 + w)) = 0 ((0.05 + 0.25w) + w(0.2 + w)) = 0 (0.05 + 0.25w + (0.2 * w + w * w)) = 0 (0.05 + 0.25w + (0.2w + w2)) = 0 Combine like terms: 0.25w + 0.2w = 0.45w (0.05 + 0.45w + w2) = 0 Solving 0.05 + 0.45w + w2 = 0 Solving for variable 'w'. Begin completing the square. Move the constant term to the right: Add '-0.05' to each side of the equation. 0.05 + 0.45w + -0.05 + w2 = 0 + -0.05 Reorder the terms: 0.05 + -0.05 + 0.45w + w2 = 0 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + 0.45w + w2 = 0 + -0.05 0.45w + w2 = 0 + -0.05 Combine like terms: 0 + -0.05 = -0.05 0.45w + w2 = -0.05 The w term is 0.45w. Take half its coefficient (0.225). Square it (0.050625) and add it to both sides. Add '0.050625' to each side of the equation. 0.45w + 0.050625 + w2 = -0.05 + 0.050625 Reorder the terms: 0.050625 + 0.45w + w2 = -0.05 + 0.050625 Combine like terms: -0.05 + 0.050625 = 0.000625 0.050625 + 0.45w + w2 = 0.000625 Factor a perfect square on the left side: (w + 0.225)(w + 0.225) = 0.000625 Calculate the square root of the right side: 0.025 Break this problem into two subproblems by setting (w + 0.225) equal to 0.025 and -0.025.Subproblem 1
w + 0.225 = 0.025 Simplifying w + 0.225 = 0.025 Reorder the terms: 0.225 + w = 0.025 Solving 0.225 + w = 0.025 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.225' to each side of the equation. 0.225 + -0.225 + w = 0.025 + -0.225 Combine like terms: 0.225 + -0.225 = 0.000 0.000 + w = 0.025 + -0.225 w = 0.025 + -0.225 Combine like terms: 0.025 + -0.225 = -0.2 w = -0.2 Simplifying w = -0.2Subproblem 2
w + 0.225 = -0.025 Simplifying w + 0.225 = -0.025 Reorder the terms: 0.225 + w = -0.025 Solving 0.225 + w = -0.025 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.225' to each side of the equation. 0.225 + -0.225 + w = -0.025 + -0.225 Combine like terms: 0.225 + -0.225 = 0.000 0.000 + w = -0.025 + -0.225 w = -0.025 + -0.225 Combine like terms: -0.025 + -0.225 = -0.25 w = -0.25 Simplifying w = -0.25Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.2, -0.25}
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