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Simplifying 2w2 + -8w + 42 + -1w2 + -7w + -14 = 0 Reorder the terms: 42 + -14 + -8w + -7w + 2w2 + -1w2 = 0 Combine like terms: 42 + -14 = 28 28 + -8w + -7w + 2w2 + -1w2 = 0 Combine like terms: -8w + -7w = -15w 28 + -15w + 2w2 + -1w2 = 0 Combine like terms: 2w2 + -1w2 = 1w2 28 + -15w + 1w2 = 0 Solving 28 + -15w + 1w2 = 0 Solving for variable 'w'. Begin completing the square. Move the constant term to the right: Add '-28' to each side of the equation. 28 + -15w + -28 + w2 = 0 + -28 Reorder the terms: 28 + -28 + -15w + w2 = 0 + -28 Combine like terms: 28 + -28 = 0 0 + -15w + w2 = 0 + -28 -15w + w2 = 0 + -28 Combine like terms: 0 + -28 = -28 -15w + w2 = -28 The w term is -15w. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15w + 56.25 + w2 = -28 + 56.25 Reorder the terms: 56.25 + -15w + w2 = -28 + 56.25 Combine like terms: -28 + 56.25 = 28.25 56.25 + -15w + w2 = 28.25 Factor a perfect square on the left side: (w + -7.5)(w + -7.5) = 28.25 Calculate the square root of the right side: 5.315072906 Break this problem into two subproblems by setting (w + -7.5) equal to 5.315072906 and -5.315072906.Subproblem 1
w + -7.5 = 5.315072906 Simplifying w + -7.5 = 5.315072906 Reorder the terms: -7.5 + w = 5.315072906 Solving -7.5 + w = 5.315072906 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + w = 5.315072906 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + w = 5.315072906 + 7.5 w = 5.315072906 + 7.5 Combine like terms: 5.315072906 + 7.5 = 12.815072906 w = 12.815072906 Simplifying w = 12.815072906Subproblem 2
w + -7.5 = -5.315072906 Simplifying w + -7.5 = -5.315072906 Reorder the terms: -7.5 + w = -5.315072906 Solving -7.5 + w = -5.315072906 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + w = -5.315072906 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + w = -5.315072906 + 7.5 w = -5.315072906 + 7.5 Combine like terms: -5.315072906 + 7.5 = 2.184927094 w = 2.184927094 Simplifying w = 2.184927094Solution
The solution to the problem is based on the solutions from the subproblems. w = {12.815072906, 2.184927094}
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