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Simplifying (2w + -2)(w) = 65 Reorder the terms: (-2 + 2w)(w) = 65 Reorder the terms for easier multiplication: w(-2 + 2w) = 65 (-2 * w + 2w * w) = 65 (-2w + 2w2) = 65 Solving -2w + 2w2 = 65 Solving for variable 'w'. Reorder the terms: -65 + -2w + 2w2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + -2w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -32.5 + -1w + w2 = 0 Move the constant term to the right: Add '32.5' to each side of the equation. -32.5 + -1w + 32.5 + w2 = 0 + 32.5 Reorder the terms: -32.5 + 32.5 + -1w + w2 = 0 + 32.5 Combine like terms: -32.5 + 32.5 = 0.0 0.0 + -1w + w2 = 0 + 32.5 -1w + w2 = 0 + 32.5 Combine like terms: 0 + 32.5 = 32.5 -1w + w2 = 32.5 The w term is -1w. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1w + 0.25 + w2 = 32.5 + 0.25 Reorder the terms: 0.25 + -1w + w2 = 32.5 + 0.25 Combine like terms: 32.5 + 0.25 = 32.75 0.25 + -1w + w2 = 32.75 Factor a perfect square on the left side: (w + -0.5)(w + -0.5) = 32.75 Calculate the square root of the right side: 5.722761571 Break this problem into two subproblems by setting (w + -0.5) equal to 5.722761571 and -5.722761571.Subproblem 1
w + -0.5 = 5.722761571 Simplifying w + -0.5 = 5.722761571 Reorder the terms: -0.5 + w = 5.722761571 Solving -0.5 + w = 5.722761571 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + w = 5.722761571 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = 5.722761571 + 0.5 w = 5.722761571 + 0.5 Combine like terms: 5.722761571 + 0.5 = 6.222761571 w = 6.222761571 Simplifying w = 6.222761571Subproblem 2
w + -0.5 = -5.722761571 Simplifying w + -0.5 = -5.722761571 Reorder the terms: -0.5 + w = -5.722761571 Solving -0.5 + w = -5.722761571 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + w = -5.722761571 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = -5.722761571 + 0.5 w = -5.722761571 + 0.5 Combine like terms: -5.722761571 + 0.5 = -5.222761571 w = -5.222761571 Simplifying w = -5.222761571Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.222761571, -5.222761571}
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