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Simplifying (2w + 3)(w) = 2500 Reorder the terms: (3 + 2w)(w) = 2500 Reorder the terms for easier multiplication: w(3 + 2w) = 2500 (3 * w + 2w * w) = 2500 (3w + 2w2) = 2500 Solving 3w + 2w2 = 2500 Solving for variable 'w'. Reorder the terms: -2500 + 3w + 2w2 = 2500 + -2500 Combine like terms: 2500 + -2500 = 0 -2500 + 3w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1250 + 1.5w + w2 = 0 Move the constant term to the right: Add '1250' to each side of the equation. -1250 + 1.5w + 1250 + w2 = 0 + 1250 Reorder the terms: -1250 + 1250 + 1.5w + w2 = 0 + 1250 Combine like terms: -1250 + 1250 = 0 0 + 1.5w + w2 = 0 + 1250 1.5w + w2 = 0 + 1250 Combine like terms: 0 + 1250 = 1250 1.5w + w2 = 1250 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 = 1250 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 1250 + 0.5625 Combine like terms: 1250 + 0.5625 = 1250.5625 0.5625 + 1.5w + w2 = 1250.5625 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 1250.5625 Calculate the square root of the right side: 35.363293116 Break this problem into two subproblems by setting (w + 0.75) equal to 35.363293116 and -35.363293116.Subproblem 1
w + 0.75 = 35.363293116 Simplifying w + 0.75 = 35.363293116 Reorder the terms: 0.75 + w = 35.363293116 Solving 0.75 + w = 35.363293116 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 = 35.363293116 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 35.363293116 + -0.75 w = 35.363293116 + -0.75 Combine like terms: 35.363293116 + -0.75 = 34.613293116 w = 34.613293116 Simplifying w = 34.613293116Subproblem 2
w + 0.75 = -35.363293116 Simplifying w + 0.75 = -35.363293116 Reorder the terms: 0.75 + w = -35.363293116 Solving 0.75 + w = -35.363293116 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 = -35.363293116 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -35.363293116 + -0.75 w = -35.363293116 + -0.75 Combine like terms: -35.363293116 + -0.75 = -36.113293116 w = -36.113293116 Simplifying w = -36.113293116Solution
The solution to the problem is based on the solutions from the subproblems. w = {34.613293116, -36.113293116}
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