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Simplifying 2w2 + -8w + 1 = w2 + 25 Reorder the terms: 1 + -8w + 2w2 = w2 + 25 Reorder the terms: 1 + -8w + 2w2 = 25 + w2 Solving 1 + -8w + 2w2 = 25 + w2 Solving for variable 'w'. Reorder the terms: 1 + -25 + -8w + 2w2 + -1w2 = 25 + w2 + -25 + -1w2 Combine like terms: 1 + -25 = -24 -24 + -8w + 2w2 + -1w2 = 25 + w2 + -25 + -1w2 Combine like terms: 2w2 + -1w2 = 1w2 -24 + -8w + 1w2 = 25 + w2 + -25 + -1w2 Reorder the terms: -24 + -8w + 1w2 = 25 + -25 + w2 + -1w2 Combine like terms: 25 + -25 = 0 -24 + -8w + 1w2 = 0 + w2 + -1w2 -24 + -8w + 1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 -24 + -8w + 1w2 = 0 Begin completing the square. Move the constant term to the right: Add '24' to each side of the equation. -24 + -8w + 24 + w2 = 0 + 24 Reorder the terms: -24 + 24 + -8w + w2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -8w + w2 = 0 + 24 -8w + w2 = 0 + 24 Combine like terms: 0 + 24 = 24 -8w + w2 = 24 The w term is -8w. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8w + 16 + w2 = 24 + 16 Reorder the terms: 16 + -8w + w2 = 24 + 16 Combine like terms: 24 + 16 = 40 16 + -8w + w2 = 40 Factor a perfect square on the left side: (w + -4)(w + -4) = 40 Calculate the square root of the right side: 6.32455532 Break this problem into two subproblems by setting (w + -4) equal to 6.32455532 and -6.32455532.Subproblem 1
w + -4 = 6.32455532 Simplifying w + -4 = 6.32455532 Reorder the terms: -4 + w = 6.32455532 Solving -4 + w = 6.32455532 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + w = 6.32455532 + 4 Combine like terms: -4 + 4 = 0 0 + w = 6.32455532 + 4 w = 6.32455532 + 4 Combine like terms: 6.32455532 + 4 = 10.32455532 w = 10.32455532 Simplifying w = 10.32455532Subproblem 2
w + -4 = -6.32455532 Simplifying w + -4 = -6.32455532 Reorder the terms: -4 + w = -6.32455532 Solving -4 + w = -6.32455532 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + w = -6.32455532 + 4 Combine like terms: -4 + 4 = 0 0 + w = -6.32455532 + 4 w = -6.32455532 + 4 Combine like terms: -6.32455532 + 4 = -2.32455532 w = -2.32455532 Simplifying w = -2.32455532Solution
The solution to the problem is based on the solutions from the subproblems. w = {10.32455532, -2.32455532}
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