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Simplifying (2w + -17.5)(w + -2.5) = 10.5 Reorder the terms: (-17.5 + 2w)(w + -2.5) = 10.5 Reorder the terms: (-17.5 + 2w)(-2.5 + w) = 10.5 Multiply (-17.5 + 2w) * (-2.5 + w) (-17.5(-2.5 + w) + 2w * (-2.5 + w)) = 10.5 ((-2.5 * -17.5 + w * -17.5) + 2w * (-2.5 + w)) = 10.5 ((43.75 + -17.5w) + 2w * (-2.5 + w)) = 10.5 (43.75 + -17.5w + (-2.5 * 2w + w * 2w)) = 10.5 (43.75 + -17.5w + (-5w + 2w2)) = 10.5 Combine like terms: -17.5w + -5w = -22.5w (43.75 + -22.5w + 2w2) = 10.5 Solving 43.75 + -22.5w + 2w2 = 10.5 Solving for variable 'w'. Reorder the terms: 43.75 + -10.5 + -22.5w + 2w2 = 10.5 + -10.5 Combine like terms: 43.75 + -10.5 = 33.25 33.25 + -22.5w + 2w2 = 10.5 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 33.25 + -22.5w + 2w2 = 0.0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 16.625 + -11.25w + w2 = 0.0 Move the constant term to the right: Add '-16.625' to each side of the equation. 16.625 + -11.25w + -16.625 + w2 = 0.0 + -16.625 Reorder the terms: 16.625 + -16.625 + -11.25w + w2 = 0.0 + -16.625 Combine like terms: 16.625 + -16.625 = 0.000 0.000 + -11.25w + w2 = 0.0 + -16.625 -11.25w + w2 = 0.0 + -16.625 Combine like terms: 0.0 + -16.625 = -16.625 -11.25w + w2 = -16.625 The w term is -11.25w. Take half its coefficient (-5.625). Square it (31.640625) and add it to both sides. Add '31.640625' to each side of the equation. -11.25w + 31.640625 + w2 = -16.625 + 31.640625 Reorder the terms: 31.640625 + -11.25w + w2 = -16.625 + 31.640625 Combine like terms: -16.625 + 31.640625 = 15.015625 31.640625 + -11.25w + w2 = 15.015625 Factor a perfect square on the left side: (w + -5.625)(w + -5.625) = 15.015625 Calculate the square root of the right side: 3.875 Break this problem into two subproblems by setting (w + -5.625) equal to 3.875 and -3.875.Subproblem 1
w + -5.625 = 3.875 Simplifying w + -5.625 = 3.875 Reorder the terms: -5.625 + w = 3.875 Solving -5.625 + w = 3.875 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.625' to each side of the equation. -5.625 + 5.625 + w = 3.875 + 5.625 Combine like terms: -5.625 + 5.625 = 0.000 0.000 + w = 3.875 + 5.625 w = 3.875 + 5.625 Combine like terms: 3.875 + 5.625 = 9.5 w = 9.5 Simplifying w = 9.5Subproblem 2
w + -5.625 = -3.875 Simplifying w + -5.625 = -3.875 Reorder the terms: -5.625 + w = -3.875 Solving -5.625 + w = -3.875 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '5.625' to each side of the equation. -5.625 + 5.625 + w = -3.875 + 5.625 Combine like terms: -5.625 + 5.625 = 0.000 0.000 + w = -3.875 + 5.625 w = -3.875 + 5.625 Combine like terms: -3.875 + 5.625 = 1.75 w = 1.75 Simplifying w = 1.75Solution
The solution to the problem is based on the solutions from the subproblems. w = {9.5, 1.75}
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