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Simplifying (2w + 6) * w = 84 Reorder the terms: (6 + 2w) * w = 84 Reorder the terms for easier multiplication: w(6 + 2w) = 84 (6 * w + 2w * w) = 84 (6w + 2w2) = 84 Solving 6w + 2w2 = 84 Solving for variable 'w'. Reorder the terms: -84 + 6w + 2w2 = 84 + -84 Combine like terms: 84 + -84 = 0 -84 + 6w + 2w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-42 + 3w + w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-42 + 3w + w2)' equal to zero and attempt to solve: Simplifying -42 + 3w + w2 = 0 Solving -42 + 3w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '42' to each side of the equation. -42 + 3w + 42 + w2 = 0 + 42 Reorder the terms: -42 + 42 + 3w + w2 = 0 + 42 Combine like terms: -42 + 42 = 0 0 + 3w + w2 = 0 + 42 3w + w2 = 0 + 42 Combine like terms: 0 + 42 = 42 3w + w2 = 42 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 = 42 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 42 + 2.25 Combine like terms: 42 + 2.25 = 44.25 2.25 + 3w + w2 = 44.25 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 44.25 Calculate the square root of the right side: 6.652067348 Break this problem into two subproblems by setting (w + 1.5) equal to 6.652067348 and -6.652067348.Subproblem 1
w + 1.5 = 6.652067348 Simplifying w + 1.5 = 6.652067348 Reorder the terms: 1.5 + w = 6.652067348 Solving 1.5 + w = 6.652067348 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 = 6.652067348 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 6.652067348 + -1.5 w = 6.652067348 + -1.5 Combine like terms: 6.652067348 + -1.5 = 5.152067348 w = 5.152067348 Simplifying w = 5.152067348Subproblem 2
w + 1.5 = -6.652067348 Simplifying w + 1.5 = -6.652067348 Reorder the terms: 1.5 + w = -6.652067348 Solving 1.5 + w = -6.652067348 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 = -6.652067348 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -6.652067348 + -1.5 w = -6.652067348 + -1.5 Combine like terms: -6.652067348 + -1.5 = -8.152067348 w = -8.152067348 Simplifying w = -8.152067348Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.152067348, -8.152067348}Solution
w = {5.152067348, -8.152067348}
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