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Simplifying (2w + 10)(w + -2) = 190 Reorder the terms: (10 + 2w)(w + -2) = 190 Reorder the terms: (10 + 2w)(-2 + w) = 190 Multiply (10 + 2w) * (-2 + w) (10(-2 + w) + 2w * (-2 + w)) = 190 ((-2 * 10 + w * 10) + 2w * (-2 + w)) = 190 ((-20 + 10w) + 2w * (-2 + w)) = 190 (-20 + 10w + (-2 * 2w + w * 2w)) = 190 (-20 + 10w + (-4w + 2w2)) = 190 Combine like terms: 10w + -4w = 6w (-20 + 6w + 2w2) = 190 Solving -20 + 6w + 2w2 = 190 Solving for variable 'w'. Reorder the terms: -20 + -190 + 6w + 2w2 = 190 + -190 Combine like terms: -20 + -190 = -210 -210 + 6w + 2w2 = 190 + -190 Combine like terms: 190 + -190 = 0 -210 + 6w + 2w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-105 + 3w + w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-105 + 3w + w2)' equal to zero and attempt to solve: Simplifying -105 + 3w + w2 = 0 Solving -105 + 3w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '105' to each side of the equation. -105 + 3w + 105 + w2 = 0 + 105 Reorder the terms: -105 + 105 + 3w + w2 = 0 + 105 Combine like terms: -105 + 105 = 0 0 + 3w + w2 = 0 + 105 3w + w2 = 0 + 105 Combine like terms: 0 + 105 = 105 3w + w2 = 105 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 = 105 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 105 + 2.25 Combine like terms: 105 + 2.25 = 107.25 2.25 + 3w + w2 = 107.25 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 107.25 Calculate the square root of the right side: 10.356157589 Break this problem into two subproblems by setting (w + 1.5) equal to 10.356157589 and -10.356157589.Subproblem 1
w + 1.5 = 10.356157589 Simplifying w + 1.5 = 10.356157589 Reorder the terms: 1.5 + w = 10.356157589 Solving 1.5 + w = 10.356157589 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 = 10.356157589 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 10.356157589 + -1.5 w = 10.356157589 + -1.5 Combine like terms: 10.356157589 + -1.5 = 8.856157589 w = 8.856157589 Simplifying w = 8.856157589Subproblem 2
w + 1.5 = -10.356157589 Simplifying w + 1.5 = -10.356157589 Reorder the terms: 1.5 + w = -10.356157589 Solving 1.5 + w = -10.356157589 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 = -10.356157589 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -10.356157589 + -1.5 w = -10.356157589 + -1.5 Combine like terms: -10.356157589 + -1.5 = -11.856157589 w = -11.856157589 Simplifying w = -11.856157589Solution
The solution to the problem is based on the solutions from the subproblems. w = {8.856157589, -11.856157589}Solution
w = {8.856157589, -11.856157589}
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