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Simplifying (w + -3) * 2(w + 7) = 114 Reorder the terms: (-3 + w) * 2(w + 7) = 114 Reorder the terms: (-3 + w) * 2(7 + w) = 114 Reorder the terms for easier multiplication: 2(-3 + w)(7 + w) = 114 Multiply (-3 + w) * (7 + w) 2(-3(7 + w) + w(7 + w)) = 114 2((7 * -3 + w * -3) + w(7 + w)) = 114 2((-21 + -3w) + w(7 + w)) = 114 2(-21 + -3w + (7 * w + w * w)) = 114 2(-21 + -3w + (7w + w2)) = 114 Combine like terms: -3w + 7w = 4w 2(-21 + 4w + w2) = 114 (-21 * 2 + 4w * 2 + w2 * 2) = 114 (-42 + 8w + 2w2) = 114 Solving -42 + 8w + 2w2 = 114 Solving for variable 'w'. Reorder the terms: -42 + -114 + 8w + 2w2 = 114 + -114 Combine like terms: -42 + -114 = -156 -156 + 8w + 2w2 = 114 + -114 Combine like terms: 114 + -114 = 0 -156 + 8w + 2w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-78 + 4w + w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-78 + 4w + w2)' equal to zero and attempt to solve: Simplifying -78 + 4w + w2 = 0 Solving -78 + 4w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + 4w + 78 + w2 = 0 + 78 Reorder the terms: -78 + 78 + 4w + w2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 4w + w2 = 0 + 78 4w + w2 = 0 + 78 Combine like terms: 0 + 78 = 78 4w + w2 = 78 The w term is 4w. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4w + 4 + w2 = 78 + 4 Reorder the terms: 4 + 4w + w2 = 78 + 4 Combine like terms: 78 + 4 = 82 4 + 4w + w2 = 82 Factor a perfect square on the left side: (w + 2)(w + 2) = 82 Calculate the square root of the right side: 9.055385138 Break this problem into two subproblems by setting (w + 2) equal to 9.055385138 and -9.055385138.Subproblem 1
w + 2 = 9.055385138 Simplifying w + 2 = 9.055385138 Reorder the terms: 2 + w = 9.055385138 Solving 2 + w = 9.055385138 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + w = 9.055385138 + -2 Combine like terms: 2 + -2 = 0 0 + w = 9.055385138 + -2 w = 9.055385138 + -2 Combine like terms: 9.055385138 + -2 = 7.055385138 w = 7.055385138 Simplifying w = 7.055385138Subproblem 2
w + 2 = -9.055385138 Simplifying w + 2 = -9.055385138 Reorder the terms: 2 + w = -9.055385138 Solving 2 + w = -9.055385138 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + w = -9.055385138 + -2 Combine like terms: 2 + -2 = 0 0 + w = -9.055385138 + -2 w = -9.055385138 + -2 Combine like terms: -9.055385138 + -2 = -11.055385138 w = -11.055385138 Simplifying w = -11.055385138Solution
The solution to the problem is based on the solutions from the subproblems. w = {7.055385138, -11.055385138}Solution
w = {7.055385138, -11.055385138}
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