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Simplifying (w + 4)(w + 4) + 3 = 23 Reorder the terms: (4 + w)(w + 4) + 3 = 23 Reorder the terms: (4 + w)(4 + w) + 3 = 23 Multiply (4 + w) * (4 + w) (4(4 + w) + w(4 + w)) + 3 = 23 ((4 * 4 + w * 4) + w(4 + w)) + 3 = 23 ((16 + 4w) + w(4 + w)) + 3 = 23 (16 + 4w + (4 * w + w * w)) + 3 = 23 (16 + 4w + (4w + w2)) + 3 = 23 Combine like terms: 4w + 4w = 8w (16 + 8w + w2) + 3 = 23 Reorder the terms: 16 + 3 + 8w + w2 = 23 Combine like terms: 16 + 3 = 19 19 + 8w + w2 = 23 Solving 19 + 8w + w2 = 23 Solving for variable 'w'. Reorder the terms: 19 + -23 + 8w + w2 = 23 + -23 Combine like terms: 19 + -23 = -4 -4 + 8w + w2 = 23 + -23 Combine like terms: 23 + -23 = 0 -4 + 8w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 8w + 4 + w2 = 0 + 4 Reorder the terms: -4 + 4 + 8w + w2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 8w + w2 = 0 + 4 8w + w2 = 0 + 4 Combine like terms: 0 + 4 = 4 8w + w2 = 4 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 = 4 + 16 Reorder the terms: 16 + 8w + w2 = 4 + 16 Combine like terms: 4 + 16 = 20 16 + 8w + w2 = 20 Factor a perfect square on the left side: (w + 4)(w + 4) = 20 Calculate the square root of the right side: 4.472135955 Break this problem into two subproblems by setting (w + 4) equal to 4.472135955 and -4.472135955.Subproblem 1
w + 4 = 4.472135955 Simplifying w + 4 = 4.472135955 Reorder the terms: 4 + w = 4.472135955 Solving 4 + w = 4.472135955 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 = 4.472135955 + -4 Combine like terms: 4 + -4 = 0 0 + w = 4.472135955 + -4 w = 4.472135955 + -4 Combine like terms: 4.472135955 + -4 = 0.472135955 w = 0.472135955 Simplifying w = 0.472135955Subproblem 2
w + 4 = -4.472135955 Simplifying w + 4 = -4.472135955 Reorder the terms: 4 + w = -4.472135955 Solving 4 + w = -4.472135955 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 = -4.472135955 + -4 Combine like terms: 4 + -4 = 0 0 + w = -4.472135955 + -4 w = -4.472135955 + -4 Combine like terms: -4.472135955 + -4 = -8.472135955 w = -8.472135955 Simplifying w = -8.472135955Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.472135955, -8.472135955}
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