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Simplifying 180 = w(30 + w) 180 = (30 * w + w * w) 180 = (30w + w2) Solving 180 = 30w + w2 Solving for variable 'w'. Reorder the terms: 180 + -30w + -1w2 = 30w + -30w + w2 + -1w2 Combine like terms: 30w + -30w = 0 180 + -30w + -1w2 = 0 + w2 + -1w2 180 + -30w + -1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 180 + -30w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -180 + 30w + w2 = 0 Move the constant term to the right: Add '180' to each side of the equation. -180 + 30w + 180 + w2 = 0 + 180 Reorder the terms: -180 + 180 + 30w + w2 = 0 + 180 Combine like terms: -180 + 180 = 0 0 + 30w + w2 = 0 + 180 30w + w2 = 0 + 180 Combine like terms: 0 + 180 = 180 30w + w2 = 180 The w term is 30w. Take half its coefficient (15). Square it (225) and add it to both sides. Add '225' to each side of the equation. 30w + 225 + w2 = 180 + 225 Reorder the terms: 225 + 30w + w2 = 180 + 225 Combine like terms: 180 + 225 = 405 225 + 30w + w2 = 405 Factor a perfect square on the left side: (w + 15)(w + 15) = 405 Calculate the square root of the right side: 20.124611797 Break this problem into two subproblems by setting (w + 15) equal to 20.124611797 and -20.124611797.Subproblem 1
w + 15 = 20.124611797 Simplifying w + 15 = 20.124611797 Reorder the terms: 15 + w = 20.124611797 Solving 15 + w = 20.124611797 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + w = 20.124611797 + -15 Combine like terms: 15 + -15 = 0 0 + w = 20.124611797 + -15 w = 20.124611797 + -15 Combine like terms: 20.124611797 + -15 = 5.124611797 w = 5.124611797 Simplifying w = 5.124611797Subproblem 2
w + 15 = -20.124611797 Simplifying w + 15 = -20.124611797 Reorder the terms: 15 + w = -20.124611797 Solving 15 + w = -20.124611797 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + w = -20.124611797 + -15 Combine like terms: 15 + -15 = 0 0 + w = -20.124611797 + -15 w = -20.124611797 + -15 Combine like terms: -20.124611797 + -15 = -35.124611797 w = -35.124611797 Simplifying w = -35.124611797Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.124611797, -35.124611797}
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