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Simplifying 2w2 + 19w + 46 = w2 + 36 Reorder the terms: 46 + 19w + 2w2 = w2 + 36 Reorder the terms: 46 + 19w + 2w2 = 36 + w2 Solving 46 + 19w + 2w2 = 36 + w2 Solving for variable 'w'. Reorder the terms: 46 + -36 + 19w + 2w2 + -1w2 = 36 + w2 + -36 + -1w2 Combine like terms: 46 + -36 = 10 10 + 19w + 2w2 + -1w2 = 36 + w2 + -36 + -1w2 Combine like terms: 2w2 + -1w2 = 1w2 10 + 19w + 1w2 = 36 + w2 + -36 + -1w2 Reorder the terms: 10 + 19w + 1w2 = 36 + -36 + w2 + -1w2 Combine like terms: 36 + -36 = 0 10 + 19w + 1w2 = 0 + w2 + -1w2 10 + 19w + 1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 10 + 19w + 1w2 = 0 Begin completing the square. Move the constant term to the right: Add '-10' to each side of the equation. 10 + 19w + -10 + w2 = 0 + -10 Reorder the terms: 10 + -10 + 19w + w2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + 19w + w2 = 0 + -10 19w + w2 = 0 + -10 Combine like terms: 0 + -10 = -10 19w + w2 = -10 The w term is 19w. Take half its coefficient (9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. 19w + 90.25 + w2 = -10 + 90.25 Reorder the terms: 90.25 + 19w + w2 = -10 + 90.25 Combine like terms: -10 + 90.25 = 80.25 90.25 + 19w + w2 = 80.25 Factor a perfect square on the left side: (w + 9.5)(w + 9.5) = 80.25 Calculate the square root of the right side: 8.958236434 Break this problem into two subproblems by setting (w + 9.5) equal to 8.958236434 and -8.958236434.Subproblem 1
w + 9.5 = 8.958236434 Simplifying w + 9.5 = 8.958236434 Reorder the terms: 9.5 + w = 8.958236434 Solving 9.5 + w = 8.958236434 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + w = 8.958236434 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + w = 8.958236434 + -9.5 w = 8.958236434 + -9.5 Combine like terms: 8.958236434 + -9.5 = -0.541763566 w = -0.541763566 Simplifying w = -0.541763566Subproblem 2
w + 9.5 = -8.958236434 Simplifying w + 9.5 = -8.958236434 Reorder the terms: 9.5 + w = -8.958236434 Solving 9.5 + w = -8.958236434 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + w = -8.958236434 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + w = -8.958236434 + -9.5 w = -8.958236434 + -9.5 Combine like terms: -8.958236434 + -9.5 = -18.458236434 w = -18.458236434 Simplifying w = -18.458236434Solution
The solution to the problem is based on the solutions from the subproblems. w = {-0.541763566, -18.458236434}
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