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Simplifying (w + -6) * 2 = 2w2 + -22w + 57 Reorder the terms: (-6 + w) * 2 = 2w2 + -22w + 57 Reorder the terms for easier multiplication: 2(-6 + w) = 2w2 + -22w + 57 (-6 * 2 + w * 2) = 2w2 + -22w + 57 (-12 + 2w) = 2w2 + -22w + 57 Reorder the terms: -12 + 2w = 57 + -22w + 2w2 Solving -12 + 2w = 57 + -22w + 2w2 Solving for variable 'w'. Reorder the terms: -12 + -57 + 2w + 22w + -2w2 = 57 + -22w + 2w2 + -57 + 22w + -2w2 Combine like terms: -12 + -57 = -69 -69 + 2w + 22w + -2w2 = 57 + -22w + 2w2 + -57 + 22w + -2w2 Combine like terms: 2w + 22w = 24w -69 + 24w + -2w2 = 57 + -22w + 2w2 + -57 + 22w + -2w2 Reorder the terms: -69 + 24w + -2w2 = 57 + -57 + -22w + 22w + 2w2 + -2w2 Combine like terms: 57 + -57 = 0 -69 + 24w + -2w2 = 0 + -22w + 22w + 2w2 + -2w2 -69 + 24w + -2w2 = -22w + 22w + 2w2 + -2w2 Combine like terms: -22w + 22w = 0 -69 + 24w + -2w2 = 0 + 2w2 + -2w2 -69 + 24w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 -69 + 24w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 34.5 + -12w + w2 = 0 Move the constant term to the right: Add '-34.5' to each side of the equation. 34.5 + -12w + -34.5 + w2 = 0 + -34.5 Reorder the terms: 34.5 + -34.5 + -12w + w2 = 0 + -34.5 Combine like terms: 34.5 + -34.5 = 0.0 0.0 + -12w + w2 = 0 + -34.5 -12w + w2 = 0 + -34.5 Combine like terms: 0 + -34.5 = -34.5 -12w + w2 = -34.5 The w term is -12w. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12w + 36 + w2 = -34.5 + 36 Reorder the terms: 36 + -12w + w2 = -34.5 + 36 Combine like terms: -34.5 + 36 = 1.5 36 + -12w + w2 = 1.5 Factor a perfect square on the left side: (w + -6)(w + -6) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (w + -6) equal to 1.224744871 and -1.224744871.Subproblem 1
w + -6 = 1.224744871 Simplifying w + -6 = 1.224744871 Reorder the terms: -6 + w = 1.224744871 Solving -6 + w = 1.224744871 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + w = 1.224744871 + 6 Combine like terms: -6 + 6 = 0 0 + w = 1.224744871 + 6 w = 1.224744871 + 6 Combine like terms: 1.224744871 + 6 = 7.224744871 w = 7.224744871 Simplifying w = 7.224744871Subproblem 2
w + -6 = -1.224744871 Simplifying w + -6 = -1.224744871 Reorder the terms: -6 + w = -1.224744871 Solving -6 + w = -1.224744871 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + w = -1.224744871 + 6 Combine like terms: -6 + 6 = 0 0 + w = -1.224744871 + 6 w = -1.224744871 + 6 Combine like terms: -1.224744871 + 6 = 4.775255129 w = 4.775255129 Simplifying w = 4.775255129Solution
The solution to the problem is based on the solutions from the subproblems. w = {7.224744871, 4.775255129}
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