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Simplifying 0.12y + 0.05y(y + 3000) = 830 Reorder the terms: 0.12y + 0.05y(3000 + y) = 830 0.12y + (3000 * 0.05y + y * 0.05y) = 830 0.12y + (150y + 0.05y2) = 830 Combine like terms: 0.12y + 150y = 150.12y 150.12y + 0.05y2 = 830 Solving 150.12y + 0.05y2 = 830 Solving for variable 'y'. Reorder the terms: -830 + 150.12y + 0.05y2 = 830 + -830 Combine like terms: 830 + -830 = 0 -830 + 150.12y + 0.05y2 = 0 Begin completing the square. Divide all terms by 0.05 the coefficient of the squared term: Divide each side by '0.05'. -16600 + 3002.4y + y2 = 0 Move the constant term to the right: Add '16600' to each side of the equation. -16600 + 3002.4y + 16600 + y2 = 0 + 16600 Reorder the terms: -16600 + 16600 + 3002.4y + y2 = 0 + 16600 Combine like terms: -16600 + 16600 = 0 0 + 3002.4y + y2 = 0 + 16600 3002.4y + y2 = 0 + 16600 Combine like terms: 0 + 16600 = 16600 3002.4y + y2 = 16600 The y term is 3002.4y. Take half its coefficient (1501.2). Square it (2253601.44) and add it to both sides. Add '2253601.44' to each side of the equation. 3002.4y + 2253601.44 + y2 = 16600 + 2253601.44 Reorder the terms: 2253601.44 + 3002.4y + y2 = 16600 + 2253601.44 Combine like terms: 16600 + 2253601.44 = 2270201.44 2253601.44 + 3002.4y + y2 = 2270201.44 Factor a perfect square on the left side: (y + 1501.2)(y + 1501.2) = 2270201.44 Calculate the square root of the right side: 1506.718766061 Break this problem into two subproblems by setting (y + 1501.2) equal to 1506.718766061 and -1506.718766061.Subproblem 1
y + 1501.2 = 1506.718766061 Simplifying y + 1501.2 = 1506.718766061 Reorder the terms: 1501.2 + y = 1506.718766061 Solving 1501.2 + y = 1506.718766061 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1501.2' to each side of the equation. 1501.2 + -1501.2 + y = 1506.718766061 + -1501.2 Combine like terms: 1501.2 + -1501.2 = 0.0 0.0 + y = 1506.718766061 + -1501.2 y = 1506.718766061 + -1501.2 Combine like terms: 1506.718766061 + -1501.2 = 5.518766061 y = 5.518766061 Simplifying y = 5.518766061Subproblem 2
y + 1501.2 = -1506.718766061 Simplifying y + 1501.2 = -1506.718766061 Reorder the terms: 1501.2 + y = -1506.718766061 Solving 1501.2 + y = -1506.718766061 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1501.2' to each side of the equation. 1501.2 + -1501.2 + y = -1506.718766061 + -1501.2 Combine like terms: 1501.2 + -1501.2 = 0.0 0.0 + y = -1506.718766061 + -1501.2 y = -1506.718766061 + -1501.2 Combine like terms: -1506.718766061 + -1501.2 = -3007.918766061 y = -3007.918766061 Simplifying y = -3007.918766061Solution
The solution to the problem is based on the solutions from the subproblems. y = {5.518766061, -3007.918766061}
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