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Simplifying 1350 = (5 + w)(30 + w) Multiply (5 + w) * (30 + w) 1350 = (5(30 + w) + w(30 + w)) 1350 = ((30 * 5 + w * 5) + w(30 + w)) 1350 = ((150 + 5w) + w(30 + w)) 1350 = (150 + 5w + (30 * w + w * w)) 1350 = (150 + 5w + (30w + w2)) Combine like terms: 5w + 30w = 35w 1350 = (150 + 35w + w2) Solving 1350 = 150 + 35w + w2 Solving for variable 'w'. Combine like terms: 1350 + -150 = 1200 1200 + -35w + -1w2 = 150 + 35w + w2 + -150 + -35w + -1w2 Reorder the terms: 1200 + -35w + -1w2 = 150 + -150 + 35w + -35w + w2 + -1w2 Combine like terms: 150 + -150 = 0 1200 + -35w + -1w2 = 0 + 35w + -35w + w2 + -1w2 1200 + -35w + -1w2 = 35w + -35w + w2 + -1w2 Combine like terms: 35w + -35w = 0 1200 + -35w + -1w2 = 0 + w2 + -1w2 1200 + -35w + -1w2 = w2 + -1w2 Combine like terms: w2 + -1w2 = 0 1200 + -35w + -1w2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1200 + 35w + w2 = 0 Move the constant term to the right: Add '1200' to each side of the equation. -1200 + 35w + 1200 + w2 = 0 + 1200 Reorder the terms: -1200 + 1200 + 35w + w2 = 0 + 1200 Combine like terms: -1200 + 1200 = 0 0 + 35w + w2 = 0 + 1200 35w + w2 = 0 + 1200 Combine like terms: 0 + 1200 = 1200 35w + w2 = 1200 The w term is 35w. Take half its coefficient (17.5). Square it (306.25) and add it to both sides. Add '306.25' to each side of the equation. 35w + 306.25 + w2 = 1200 + 306.25 Reorder the terms: 306.25 + 35w + w2 = 1200 + 306.25 Combine like terms: 1200 + 306.25 = 1506.25 306.25 + 35w + w2 = 1506.25 Factor a perfect square on the left side: (w + 17.5)(w + 17.5) = 1506.25 Calculate the square root of the right side: 38.810436741 Break this problem into two subproblems by setting (w + 17.5) equal to 38.810436741 and -38.810436741.Subproblem 1
w + 17.5 = 38.810436741 Simplifying w + 17.5 = 38.810436741 Reorder the terms: 17.5 + w = 38.810436741 Solving 17.5 + w = 38.810436741 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-17.5' to each side of the equation. 17.5 + -17.5 + w = 38.810436741 + -17.5 Combine like terms: 17.5 + -17.5 = 0.0 0.0 + w = 38.810436741 + -17.5 w = 38.810436741 + -17.5 Combine like terms: 38.810436741 + -17.5 = 21.310436741 w = 21.310436741 Simplifying w = 21.310436741Subproblem 2
w + 17.5 = -38.810436741 Simplifying w + 17.5 = -38.810436741 Reorder the terms: 17.5 + w = -38.810436741 Solving 17.5 + w = -38.810436741 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-17.5' to each side of the equation. 17.5 + -17.5 + w = -38.810436741 + -17.5 Combine like terms: 17.5 + -17.5 = 0.0 0.0 + w = -38.810436741 + -17.5 w = -38.810436741 + -17.5 Combine like terms: -38.810436741 + -17.5 = -56.310436741 w = -56.310436741 Simplifying w = -56.310436741Solution
The solution to the problem is based on the solutions from the subproblems. w = {21.310436741, -56.310436741}
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