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Simplifying 1.5w(180 + -3w) = 180 (180 * 1.5w + -3w * 1.5w) = 180 (270w + -4.5w2) = 180 Solving 270w + -4.5w2 = 180 Solving for variable 'w'. Reorder the terms: -180 + 270w + -4.5w2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 270w + -4.5w2 = 0 Begin completing the square. Divide all terms by -4.5 the coefficient of the squared term: Divide each side by '-4.5'. 40 + -60w + w2 = 0 Move the constant term to the right: Add '-40' to each side of the equation. 40 + -60w + -40 + w2 = 0 + -40 Reorder the terms: 40 + -40 + -60w + w2 = 0 + -40 Combine like terms: 40 + -40 = 0 0 + -60w + w2 = 0 + -40 -60w + w2 = 0 + -40 Combine like terms: 0 + -40 = -40 -60w + w2 = -40 The w term is -60w. Take half its coefficient (-30). Square it (900) and add it to both sides. Add '900' to each side of the equation. -60w + 900 + w2 = -40 + 900 Reorder the terms: 900 + -60w + w2 = -40 + 900 Combine like terms: -40 + 900 = 860 900 + -60w + w2 = 860 Factor a perfect square on the left side: (w + -30)(w + -30) = 860 Calculate the square root of the right side: 29.325756597 Break this problem into two subproblems by setting (w + -30) equal to 29.325756597 and -29.325756597.Subproblem 1
w + -30 = 29.325756597 Simplifying w + -30 = 29.325756597 Reorder the terms: -30 + w = 29.325756597 Solving -30 + w = 29.325756597 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '30' to each side of the equation. -30 + 30 + w = 29.325756597 + 30 Combine like terms: -30 + 30 = 0 0 + w = 29.325756597 + 30 w = 29.325756597 + 30 Combine like terms: 29.325756597 + 30 = 59.325756597 w = 59.325756597 Simplifying w = 59.325756597Subproblem 2
w + -30 = -29.325756597 Simplifying w + -30 = -29.325756597 Reorder the terms: -30 + w = -29.325756597 Solving -30 + w = -29.325756597 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '30' to each side of the equation. -30 + 30 + w = -29.325756597 + 30 Combine like terms: -30 + 30 = 0 0 + w = -29.325756597 + 30 w = -29.325756597 + 30 Combine like terms: -29.325756597 + 30 = 0.674243403 w = 0.674243403 Simplifying w = 0.674243403Solution
The solution to the problem is based on the solutions from the subproblems. w = {59.325756597, 0.674243403}
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