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Simplifying (8 + 2w)(13 + 2w) = 164 Multiply (8 + 2w) * (13 + 2w) (8(13 + 2w) + 2w * (13 + 2w)) = 164 ((13 * 8 + 2w * 8) + 2w * (13 + 2w)) = 164 ((104 + 16w) + 2w * (13 + 2w)) = 164 (104 + 16w + (13 * 2w + 2w * 2w)) = 164 (104 + 16w + (26w + 4w2)) = 164 Combine like terms: 16w + 26w = 42w (104 + 42w + 4w2) = 164 Solving 104 + 42w + 4w2 = 164 Solving for variable 'w'. Reorder the terms: 104 + -164 + 42w + 4w2 = 164 + -164 Combine like terms: 104 + -164 = -60 -60 + 42w + 4w2 = 164 + -164 Combine like terms: 164 + -164 = 0 -60 + 42w + 4w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-30 + 21w + 2w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-30 + 21w + 2w2)' equal to zero and attempt to solve: Simplifying -30 + 21w + 2w2 = 0 Solving -30 + 21w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -15 + 10.5w + w2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 10.5w + 15 + w2 = 0 + 15 Reorder the terms: -15 + 15 + 10.5w + w2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 10.5w + w2 = 0 + 15 10.5w + w2 = 0 + 15 Combine like terms: 0 + 15 = 15 10.5w + w2 = 15 The w term is 10.5w. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5w + 27.5625 + w2 = 15 + 27.5625 Reorder the terms: 27.5625 + 10.5w + w2 = 15 + 27.5625 Combine like terms: 15 + 27.5625 = 42.5625 27.5625 + 10.5w + w2 = 42.5625 Factor a perfect square on the left side: (w + 5.25)(w + 5.25) = 42.5625 Calculate the square root of the right side: 6.523994175 Break this problem into two subproblems by setting (w + 5.25) equal to 6.523994175 and -6.523994175.Subproblem 1
w + 5.25 = 6.523994175 Simplifying w + 5.25 = 6.523994175 Reorder the terms: 5.25 + w = 6.523994175 Solving 5.25 + w = 6.523994175 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + w = 6.523994175 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + w = 6.523994175 + -5.25 w = 6.523994175 + -5.25 Combine like terms: 6.523994175 + -5.25 = 1.273994175 w = 1.273994175 Simplifying w = 1.273994175Subproblem 2
w + 5.25 = -6.523994175 Simplifying w + 5.25 = -6.523994175 Reorder the terms: 5.25 + w = -6.523994175 Solving 5.25 + w = -6.523994175 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + w = -6.523994175 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + w = -6.523994175 + -5.25 w = -6.523994175 + -5.25 Combine like terms: -6.523994175 + -5.25 = -11.773994175 w = -11.773994175 Simplifying w = -11.773994175Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.273994175, -11.773994175}Solution
w = {1.273994175, -11.773994175}
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