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Simplifying (2w + 18)(w + -4) + -84 = 0 Reorder the terms: (18 + 2w)(w + -4) + -84 = 0 Reorder the terms: (18 + 2w)(-4 + w) + -84 = 0 Multiply (18 + 2w) * (-4 + w) (18(-4 + w) + 2w * (-4 + w)) + -84 = 0 ((-4 * 18 + w * 18) + 2w * (-4 + w)) + -84 = 0 ((-72 + 18w) + 2w * (-4 + w)) + -84 = 0 (-72 + 18w + (-4 * 2w + w * 2w)) + -84 = 0 (-72 + 18w + (-8w + 2w2)) + -84 = 0 Combine like terms: 18w + -8w = 10w (-72 + 10w + 2w2) + -84 = 0 Reorder the terms: -72 + -84 + 10w + 2w2 = 0 Combine like terms: -72 + -84 = -156 -156 + 10w + 2w2 = 0 Solving -156 + 10w + 2w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2'. 2(-78 + 5w + w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-78 + 5w + w2)' equal to zero and attempt to solve: Simplifying -78 + 5w + w2 = 0 Solving -78 + 5w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + 5w + 78 + w2 = 0 + 78 Reorder the terms: -78 + 78 + 5w + w2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 5w + w2 = 0 + 78 5w + w2 = 0 + 78 Combine like terms: 0 + 78 = 78 5w + w2 = 78 The w term is 5w. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5w + 6.25 + w2 = 78 + 6.25 Reorder the terms: 6.25 + 5w + w2 = 78 + 6.25 Combine like terms: 78 + 6.25 = 84.25 6.25 + 5w + w2 = 84.25 Factor a perfect square on the left side: (w + 2.5)(w + 2.5) = 84.25 Calculate the square root of the right side: 9.178779875 Break this problem into two subproblems by setting (w + 2.5) equal to 9.178779875 and -9.178779875.Subproblem 1
w + 2.5 = 9.178779875 Simplifying w + 2.5 = 9.178779875 Reorder the terms: 2.5 + w = 9.178779875 Solving 2.5 + w = 9.178779875 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = 9.178779875 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = 9.178779875 + -2.5 w = 9.178779875 + -2.5 Combine like terms: 9.178779875 + -2.5 = 6.678779875 w = 6.678779875 Simplifying w = 6.678779875Subproblem 2
w + 2.5 = -9.178779875 Simplifying w + 2.5 = -9.178779875 Reorder the terms: 2.5 + w = -9.178779875 Solving 2.5 + w = -9.178779875 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = -9.178779875 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = -9.178779875 + -2.5 w = -9.178779875 + -2.5 Combine like terms: -9.178779875 + -2.5 = -11.678779875 w = -11.678779875 Simplifying w = -11.678779875Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.678779875, -11.678779875}Solution
w = {6.678779875, -11.678779875}
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