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Simplifying 0.12y + 0.08y(y + 5000) = 1400 Reorder the terms: 0.12y + 0.08y(5000 + y) = 1400 0.12y + (5000 * 0.08y + y * 0.08y) = 1400 0.12y + (400y + 0.08y2) = 1400 Combine like terms: 0.12y + 400y = 400.12y 400.12y + 0.08y2 = 1400 Solving 400.12y + 0.08y2 = 1400 Solving for variable 'y'. Reorder the terms: -1400 + 400.12y + 0.08y2 = 1400 + -1400 Combine like terms: 1400 + -1400 = 0 -1400 + 400.12y + 0.08y2 = 0 Begin completing the square. Divide all terms by 0.08 the coefficient of the squared term: Divide each side by '0.08'. -17500 + 5001.5y + y2 = 0 Move the constant term to the right: Add '17500' to each side of the equation. -17500 + 5001.5y + 17500 + y2 = 0 + 17500 Reorder the terms: -17500 + 17500 + 5001.5y + y2 = 0 + 17500 Combine like terms: -17500 + 17500 = 0 0 + 5001.5y + y2 = 0 + 17500 5001.5y + y2 = 0 + 17500 Combine like terms: 0 + 17500 = 17500 5001.5y + y2 = 17500 The y term is 5001.5y. Take half its coefficient (2500.75). Square it (6253750.563) and add it to both sides. Add '6253750.563' to each side of the equation. 5001.5y + 6253750.563 + y2 = 17500 + 6253750.563 Reorder the terms: 6253750.563 + 5001.5y + y2 = 17500 + 6253750.563 Combine like terms: 17500 + 6253750.563 = 6271250.563 6253750.563 + 5001.5y + y2 = 6271250.563 Factor a perfect square on the left side: (y + 2500.75)(y + 2500.75) = 6271250.563 Calculate the square root of the right side: 2504.246506037 Break this problem into two subproblems by setting (y + 2500.75) equal to 2504.246506037 and -2504.246506037.Subproblem 1
y + 2500.75 = 2504.246506037 Simplifying y + 2500.75 = 2504.246506037 Reorder the terms: 2500.75 + y = 2504.246506037 Solving 2500.75 + y = 2504.246506037 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2500.75' to each side of the equation. 2500.75 + -2500.75 + y = 2504.246506037 + -2500.75 Combine like terms: 2500.75 + -2500.75 = 0.00 0.00 + y = 2504.246506037 + -2500.75 y = 2504.246506037 + -2500.75 Combine like terms: 2504.246506037 + -2500.75 = 3.496506037 y = 3.496506037 Simplifying y = 3.496506037Subproblem 2
y + 2500.75 = -2504.246506037 Simplifying y + 2500.75 = -2504.246506037 Reorder the terms: 2500.75 + y = -2504.246506037 Solving 2500.75 + y = -2504.246506037 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2500.75' to each side of the equation. 2500.75 + -2500.75 + y = -2504.246506037 + -2500.75 Combine like terms: 2500.75 + -2500.75 = 0.00 0.00 + y = -2504.246506037 + -2500.75 y = -2504.246506037 + -2500.75 Combine like terms: -2504.246506037 + -2500.75 = -5004.996506037 y = -5004.996506037 Simplifying y = -5004.996506037Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.496506037, -5004.996506037}
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