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Simplifying w(2w + 3) = 200 Reorder the terms: w(3 + 2w) = 200 (3 * w + 2w * w) = 200 (3w + 2w2) = 200 Solving 3w + 2w2 = 200 Solving for variable 'w'. Reorder the terms: -200 + 3w + 2w2 = 200 + -200 Combine like terms: 200 + -200 = 0 -200 + 3w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -100 + 1.5w + w2 = 0 Move the constant term to the right: Add '100' to each side of the equation. -100 + 1.5w + 100 + w2 = 0 + 100 Reorder the terms: -100 + 100 + 1.5w + w2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + 1.5w + w2 = 0 + 100 1.5w + w2 = 0 + 100 Combine like terms: 0 + 100 = 100 1.5w + w2 = 100 The w term is 1.5w. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5w + 0.5625 + w2 = 100 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 100 + 0.5625 Combine like terms: 100 + 0.5625 = 100.5625 0.5625 + 1.5w + w2 = 100.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 100.5625 Calculate the square root of the right side: 10.02808556 Break this problem into two subproblems by setting (w + 0.75) equal to 10.02808556 and -10.02808556.Subproblem 1
w + 0.75 = 10.02808556 Simplifying w + 0.75 = 10.02808556 Reorder the terms: 0.75 + w = 10.02808556 Solving 0.75 + w = 10.02808556 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 10.02808556 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 10.02808556 + -0.75 w = 10.02808556 + -0.75 Combine like terms: 10.02808556 + -0.75 = 9.27808556 w = 9.27808556 Simplifying w = 9.27808556Subproblem 2
w + 0.75 = -10.02808556 Simplifying w + 0.75 = -10.02808556 Reorder the terms: 0.75 + w = -10.02808556 Solving 0.75 + w = -10.02808556 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -10.02808556 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -10.02808556 + -0.75 w = -10.02808556 + -0.75 Combine like terms: -10.02808556 + -0.75 = -10.77808556 w = -10.77808556 Simplifying w = -10.77808556Solution
The solution to the problem is based on the solutions from the subproblems. w = {9.27808556, -10.77808556}
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