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Simplifying (2w + 3) * w = 66 Reorder the terms: (3 + 2w) * w = 66 Reorder the terms for easier multiplication: w(3 + 2w) = 66 (3 * w + 2w * w) = 66 (3w + 2w2) = 66 Solving 3w + 2w2 = 66 Solving for variable 'w'. Reorder the terms: -66 + 3w + 2w2 = 66 + -66 Combine like terms: 66 + -66 = 0 -66 + 3w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -33 + 1.5w + w2 = 0 Move the constant term to the right: Add '33' to each side of the equation. -33 + 1.5w + 33 + w2 = 0 + 33 Reorder the terms: -33 + 33 + 1.5w + w2 = 0 + 33 Combine like terms: -33 + 33 = 0 0 + 1.5w + w2 = 0 + 33 1.5w + w2 = 0 + 33 Combine like terms: 0 + 33 = 33 1.5w + w2 = 33 The w term is 1.5w. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5w + 0.5625 + w2 = 33 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 33 + 0.5625 Combine like terms: 33 + 0.5625 = 33.5625 0.5625 + 1.5w + w2 = 33.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 33.5625 Calculate the square root of the right side: 5.793315113 Break this problem into two subproblems by setting (w + 0.75) equal to 5.793315113 and -5.793315113.Subproblem 1
w + 0.75 = 5.793315113 Simplifying w + 0.75 = 5.793315113 Reorder the terms: 0.75 + w = 5.793315113 Solving 0.75 + w = 5.793315113 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 5.793315113 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 5.793315113 + -0.75 w = 5.793315113 + -0.75 Combine like terms: 5.793315113 + -0.75 = 5.043315113 w = 5.043315113 Simplifying w = 5.043315113Subproblem 2
w + 0.75 = -5.793315113 Simplifying w + 0.75 = -5.793315113 Reorder the terms: 0.75 + w = -5.793315113 Solving 0.75 + w = -5.793315113 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -5.793315113 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -5.793315113 + -0.75 w = -5.793315113 + -0.75 Combine like terms: -5.793315113 + -0.75 = -6.543315113 w = -6.543315113 Simplifying w = -6.543315113Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.043315113, -6.543315113}
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