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Simplifying (4 + 2w) * w = 74 Reorder the terms for easier multiplication: w(4 + 2w) = 74 (4 * w + 2w * w) = 74 (4w + 2w2) = 74 Solving 4w + 2w2 = 74 Solving for variable 'w'. Reorder the terms: -74 + 4w + 2w2 = 74 + -74 Combine like terms: 74 + -74 = 0 -74 + 4w + 2w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-37 + 2w + w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-37 + 2w + w2)' equal to zero and attempt to solve: Simplifying -37 + 2w + w2 = 0 Solving -37 + 2w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '37' to each side of the equation. -37 + 2w + 37 + w2 = 0 + 37 Reorder the terms: -37 + 37 + 2w + w2 = 0 + 37 Combine like terms: -37 + 37 = 0 0 + 2w + w2 = 0 + 37 2w + w2 = 0 + 37 Combine like terms: 0 + 37 = 37 2w + w2 = 37 The w term is 2w. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2w + 1 + w2 = 37 + 1 Reorder the terms: 1 + 2w + w2 = 37 + 1 Combine like terms: 37 + 1 = 38 1 + 2w + w2 = 38 Factor a perfect square on the left side: (w + 1)(w + 1) = 38 Calculate the square root of the right side: 6.164414003 Break this problem into two subproblems by setting (w + 1) equal to 6.164414003 and -6.164414003.Subproblem 1
w + 1 = 6.164414003 Simplifying w + 1 = 6.164414003 Reorder the terms: 1 + w = 6.164414003 Solving 1 + w = 6.164414003 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + w = 6.164414003 + -1 Combine like terms: 1 + -1 = 0 0 + w = 6.164414003 + -1 w = 6.164414003 + -1 Combine like terms: 6.164414003 + -1 = 5.164414003 w = 5.164414003 Simplifying w = 5.164414003Subproblem 2
w + 1 = -6.164414003 Simplifying w + 1 = -6.164414003 Reorder the terms: 1 + w = -6.164414003 Solving 1 + w = -6.164414003 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + w = -6.164414003 + -1 Combine like terms: 1 + -1 = 0 0 + w = -6.164414003 + -1 w = -6.164414003 + -1 Combine like terms: -6.164414003 + -1 = -7.164414003 w = -7.164414003 Simplifying w = -7.164414003Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.164414003, -7.164414003}Solution
w = {5.164414003, -7.164414003}
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