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Simplifying 2(w + -5)(w + -2) = 190 Reorder the terms: 2(-5 + w)(w + -2) = 190 Reorder the terms: 2(-5 + w)(-2 + w) = 190 Multiply (-5 + w) * (-2 + w) 2(-5(-2 + w) + w(-2 + w)) = 190 2((-2 * -5 + w * -5) + w(-2 + w)) = 190 2((10 + -5w) + w(-2 + w)) = 190 2(10 + -5w + (-2 * w + w * w)) = 190 2(10 + -5w + (-2w + w2)) = 190 Combine like terms: -5w + -2w = -7w 2(10 + -7w + w2) = 190 (10 * 2 + -7w * 2 + w2 * 2) = 190 (20 + -14w + 2w2) = 190 Solving 20 + -14w + 2w2 = 190 Solving for variable 'w'. Reorder the terms: 20 + -190 + -14w + 2w2 = 190 + -190 Combine like terms: 20 + -190 = -170 -170 + -14w + 2w2 = 190 + -190 Combine like terms: 190 + -190 = 0 -170 + -14w + 2w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-85 + -7w + w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-85 + -7w + w2)' equal to zero and attempt to solve: Simplifying -85 + -7w + w2 = 0 Solving -85 + -7w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '85' to each side of the equation. -85 + -7w + 85 + w2 = 0 + 85 Reorder the terms: -85 + 85 + -7w + w2 = 0 + 85 Combine like terms: -85 + 85 = 0 0 + -7w + w2 = 0 + 85 -7w + w2 = 0 + 85 Combine like terms: 0 + 85 = 85 -7w + w2 = 85 The w term is -7w. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7w + 12.25 + w2 = 85 + 12.25 Reorder the terms: 12.25 + -7w + w2 = 85 + 12.25 Combine like terms: 85 + 12.25 = 97.25 12.25 + -7w + w2 = 97.25 Factor a perfect square on the left side: (w + -3.5)(w + -3.5) = 97.25 Calculate the square root of the right side: 9.861541462 Break this problem into two subproblems by setting (w + -3.5) equal to 9.861541462 and -9.861541462.Subproblem 1
w + -3.5 = 9.861541462 Simplifying w + -3.5 = 9.861541462 Reorder the terms: -3.5 + w = 9.861541462 Solving -3.5 + w = 9.861541462 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + w = 9.861541462 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + w = 9.861541462 + 3.5 w = 9.861541462 + 3.5 Combine like terms: 9.861541462 + 3.5 = 13.361541462 w = 13.361541462 Simplifying w = 13.361541462Subproblem 2
w + -3.5 = -9.861541462 Simplifying w + -3.5 = -9.861541462 Reorder the terms: -3.5 + w = -9.861541462 Solving -3.5 + w = -9.861541462 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + w = -9.861541462 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + w = -9.861541462 + 3.5 w = -9.861541462 + 3.5 Combine like terms: -9.861541462 + 3.5 = -6.361541462 w = -6.361541462 Simplifying w = -6.361541462Solution
The solution to the problem is based on the solutions from the subproblems. w = {13.361541462, -6.361541462}Solution
w = {13.361541462, -6.361541462}
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