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Simplifying (8 + 2w)(13 + 2w) = 163 Multiply (8 + 2w) * (13 + 2w) (8(13 + 2w) + 2w * (13 + 2w)) = 163 ((13 * 8 + 2w * 8) + 2w * (13 + 2w)) = 163 ((104 + 16w) + 2w * (13 + 2w)) = 163 (104 + 16w + (13 * 2w + 2w * 2w)) = 163 (104 + 16w + (26w + 4w2)) = 163 Combine like terms: 16w + 26w = 42w (104 + 42w + 4w2) = 163 Solving 104 + 42w + 4w2 = 163 Solving for variable 'w'. Reorder the terms: 104 + -163 + 42w + 4w2 = 163 + -163 Combine like terms: 104 + -163 = -59 -59 + 42w + 4w2 = 163 + -163 Combine like terms: 163 + -163 = 0 -59 + 42w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -14.75 + 10.5w + w2 = 0 Move the constant term to the right: Add '14.75' to each side of the equation. -14.75 + 10.5w + 14.75 + w2 = 0 + 14.75 Reorder the terms: -14.75 + 14.75 + 10.5w + w2 = 0 + 14.75 Combine like terms: -14.75 + 14.75 = 0.00 0.00 + 10.5w + w2 = 0 + 14.75 10.5w + w2 = 0 + 14.75 Combine like terms: 0 + 14.75 = 14.75 10.5w + w2 = 14.75 The w term is 10.5w. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5w + 27.5625 + w2 = 14.75 + 27.5625 Reorder the terms: 27.5625 + 10.5w + w2 = 14.75 + 27.5625 Combine like terms: 14.75 + 27.5625 = 42.3125 27.5625 + 10.5w + w2 = 42.3125 Factor a perfect square on the left side: (w + 5.25)(w + 5.25) = 42.3125 Calculate the square root of the right side: 6.504805916 Break this problem into two subproblems by setting (w + 5.25) equal to 6.504805916 and -6.504805916.Subproblem 1
w + 5.25 = 6.504805916 Simplifying w + 5.25 = 6.504805916 Reorder the terms: 5.25 + w = 6.504805916 Solving 5.25 + w = 6.504805916 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + w = 6.504805916 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + w = 6.504805916 + -5.25 w = 6.504805916 + -5.25 Combine like terms: 6.504805916 + -5.25 = 1.254805916 w = 1.254805916 Simplifying w = 1.254805916Subproblem 2
w + 5.25 = -6.504805916 Simplifying w + 5.25 = -6.504805916 Reorder the terms: 5.25 + w = -6.504805916 Solving 5.25 + w = -6.504805916 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + w = -6.504805916 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + w = -6.504805916 + -5.25 w = -6.504805916 + -5.25 Combine like terms: -6.504805916 + -5.25 = -11.754805916 w = -11.754805916 Simplifying w = -11.754805916Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.254805916, -11.754805916}
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