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Simplifying 12w + -2 + 6w + 24 + -3 = -3w2 + -6w + 15w + 35 Reorder the terms: -2 + 24 + -3 + 12w + 6w = -3w2 + -6w + 15w + 35 Combine like terms: -2 + 24 = 22 22 + -3 + 12w + 6w = -3w2 + -6w + 15w + 35 Combine like terms: 22 + -3 = 19 19 + 12w + 6w = -3w2 + -6w + 15w + 35 Combine like terms: 12w + 6w = 18w 19 + 18w = -3w2 + -6w + 15w + 35 Reorder the terms: 19 + 18w = 35 + -6w + 15w + -3w2 Combine like terms: -6w + 15w = 9w 19 + 18w = 35 + 9w + -3w2 Solving 19 + 18w = 35 + 9w + -3w2 Solving for variable 'w'. Reorder the terms: 19 + -35 + 18w + -9w + 3w2 = 35 + 9w + -3w2 + -35 + -9w + 3w2 Combine like terms: 19 + -35 = -16 -16 + 18w + -9w + 3w2 = 35 + 9w + -3w2 + -35 + -9w + 3w2 Combine like terms: 18w + -9w = 9w -16 + 9w + 3w2 = 35 + 9w + -3w2 + -35 + -9w + 3w2 Reorder the terms: -16 + 9w + 3w2 = 35 + -35 + 9w + -9w + -3w2 + 3w2 Combine like terms: 35 + -35 = 0 -16 + 9w + 3w2 = 0 + 9w + -9w + -3w2 + 3w2 -16 + 9w + 3w2 = 9w + -9w + -3w2 + 3w2 Combine like terms: 9w + -9w = 0 -16 + 9w + 3w2 = 0 + -3w2 + 3w2 -16 + 9w + 3w2 = -3w2 + 3w2 Combine like terms: -3w2 + 3w2 = 0 -16 + 9w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -5.333333333 + 3w + w2 = 0 Move the constant term to the right: Add '5.333333333' to each side of the equation. -5.333333333 + 3w + 5.333333333 + w2 = 0 + 5.333333333 Reorder the terms: -5.333333333 + 5.333333333 + 3w + w2 = 0 + 5.333333333 Combine like terms: -5.333333333 + 5.333333333 = 0.000000000 0.000000000 + 3w + w2 = 0 + 5.333333333 3w + w2 = 0 + 5.333333333 Combine like terms: 0 + 5.333333333 = 5.333333333 3w + w2 = 5.333333333 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 5.333333333 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 5.333333333 + 2.25 Combine like terms: 5.333333333 + 2.25 = 7.583333333 2.25 + 3w + w2 = 7.583333333 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 7.583333333 Calculate the square root of the right side: 2.753785274 Break this problem into two subproblems by setting (w + 1.5) equal to 2.753785274 and -2.753785274.Subproblem 1
w + 1.5 = 2.753785274 Simplifying w + 1.5 = 2.753785274 Reorder the terms: 1.5 + w = 2.753785274 Solving 1.5 + w = 2.753785274 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 2.753785274 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 2.753785274 + -1.5 w = 2.753785274 + -1.5 Combine like terms: 2.753785274 + -1.5 = 1.253785274 w = 1.253785274 Simplifying w = 1.253785274Subproblem 2
w + 1.5 = -2.753785274 Simplifying w + 1.5 = -2.753785274 Reorder the terms: 1.5 + w = -2.753785274 Solving 1.5 + w = -2.753785274 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -2.753785274 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -2.753785274 + -1.5 w = -2.753785274 + -1.5 Combine like terms: -2.753785274 + -1.5 = -4.253785274 w = -4.253785274 Simplifying w = -4.253785274Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.253785274, -4.253785274}
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