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Simplifying (9 + 2w)(w + -4) = 110 Reorder the terms: (9 + 2w)(-4 + w) = 110 Multiply (9 + 2w) * (-4 + w) (9(-4 + w) + 2w * (-4 + w)) = 110 ((-4 * 9 + w * 9) + 2w * (-4 + w)) = 110 ((-36 + 9w) + 2w * (-4 + w)) = 110 (-36 + 9w + (-4 * 2w + w * 2w)) = 110 (-36 + 9w + (-8w + 2w2)) = 110 Combine like terms: 9w + -8w = 1w (-36 + 1w + 2w2) = 110 Solving -36 + 1w + 2w2 = 110 Solving for variable 'w'. Reorder the terms: -36 + -110 + 1w + 2w2 = 110 + -110 Combine like terms: -36 + -110 = -146 -146 + 1w + 2w2 = 110 + -110 Combine like terms: 110 + -110 = 0 -146 + 1w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -73 + 0.5w + w2 = 0 Move the constant term to the right: Add '73' to each side of the equation. -73 + 0.5w + 73 + w2 = 0 + 73 Reorder the terms: -73 + 73 + 0.5w + w2 = 0 + 73 Combine like terms: -73 + 73 = 0 0 + 0.5w + w2 = 0 + 73 0.5w + w2 = 0 + 73 Combine like terms: 0 + 73 = 73 0.5w + w2 = 73 The w term is 0.5w. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5w + 0.0625 + w2 = 73 + 0.0625 Reorder the terms: 0.0625 + 0.5w + w2 = 73 + 0.0625 Combine like terms: 73 + 0.0625 = 73.0625 0.0625 + 0.5w + w2 = 73.0625 Factor a perfect square on the left side: (w + 0.25)(w + 0.25) = 73.0625 Calculate the square root of the right side: 8.547660499 Break this problem into two subproblems by setting (w + 0.25) equal to 8.547660499 and -8.547660499.Subproblem 1
w + 0.25 = 8.547660499 Simplifying w + 0.25 = 8.547660499 Reorder the terms: 0.25 + w = 8.547660499 Solving 0.25 + w = 8.547660499 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = 8.547660499 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = 8.547660499 + -0.25 w = 8.547660499 + -0.25 Combine like terms: 8.547660499 + -0.25 = 8.297660499 w = 8.297660499 Simplifying w = 8.297660499Subproblem 2
w + 0.25 = -8.547660499 Simplifying w + 0.25 = -8.547660499 Reorder the terms: 0.25 + w = -8.547660499 Solving 0.25 + w = -8.547660499 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = -8.547660499 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = -8.547660499 + -0.25 w = -8.547660499 + -0.25 Combine like terms: -8.547660499 + -0.25 = -8.797660499 w = -8.797660499 Simplifying w = -8.797660499Solution
The solution to the problem is based on the solutions from the subproblems. w = {8.297660499, -8.797660499}
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