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Simplifying 500x + 0.1x2 + 17526 = 1100x + -0.01x2 Reorder the terms: 17526 + 500x + 0.1x2 = 1100x + -0.01x2 Solving 17526 + 500x + 0.1x2 = 1100x + -0.01x2 Solving for variable 'x'. Reorder the terms: 17526 + 500x + -1100x + 0.1x2 + 0.01x2 = 1100x + -0.01x2 + -1100x + 0.01x2 Combine like terms: 500x + -1100x = -600x 17526 + -600x + 0.1x2 + 0.01x2 = 1100x + -0.01x2 + -1100x + 0.01x2 Combine like terms: 0.1x2 + 0.01x2 = 0.11x2 17526 + -600x + 0.11x2 = 1100x + -0.01x2 + -1100x + 0.01x2 Reorder the terms: 17526 + -600x + 0.11x2 = 1100x + -1100x + -0.01x2 + 0.01x2 Combine like terms: 1100x + -1100x = 0 17526 + -600x + 0.11x2 = 0 + -0.01x2 + 0.01x2 17526 + -600x + 0.11x2 = -0.01x2 + 0.01x2 Combine like terms: -0.01x2 + 0.01x2 = 0.00 17526 + -600x + 0.11x2 = 0.00 Begin completing the square. Divide all terms by 0.11 the coefficient of the squared term: Divide each side by '0.11'. 159327.2727 + -5454.545455x + x2 = 0 Move the constant term to the right: Add '-159327.2727' to each side of the equation. 159327.2727 + -5454.545455x + -159327.2727 + x2 = 0 + -159327.2727 Reorder the terms: 159327.2727 + -159327.2727 + -5454.545455x + x2 = 0 + -159327.2727 Combine like terms: 159327.2727 + -159327.2727 = 0.0000 0.0000 + -5454.545455x + x2 = 0 + -159327.2727 -5454.545455x + x2 = 0 + -159327.2727 Combine like terms: 0 + -159327.2727 = -159327.2727 -5454.545455x + x2 = -159327.2727 The x term is -5454.545455x. Take half its coefficient (-2727.272728). Square it (7438016.533) and add it to both sides. Add '7438016.533' to each side of the equation. -5454.545455x + 7438016.533 + x2 = -159327.2727 + 7438016.533 Reorder the terms: 7438016.533 + -5454.545455x + x2 = -159327.2727 + 7438016.533 Combine like terms: -159327.2727 + 7438016.533 = 7278689.2603 7438016.533 + -5454.545455x + x2 = 7278689.2603 Factor a perfect square on the left side: (x + -2727.272728)(x + -2727.272728) = 7278689.2603 Calculate the square root of the right side: 2697.904605486 Break this problem into two subproblems by setting (x + -2727.272728) equal to 2697.904605486 and -2697.904605486.Subproblem 1
x + -2727.272728 = 2697.904605486 Simplifying x + -2727.272728 = 2697.904605486 Reorder the terms: -2727.272728 + x = 2697.904605486 Solving -2727.272728 + x = 2697.904605486 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2727.272728' to each side of the equation. -2727.272728 + 2727.272728 + x = 2697.904605486 + 2727.272728 Combine like terms: -2727.272728 + 2727.272728 = 0.000000 0.000000 + x = 2697.904605486 + 2727.272728 x = 2697.904605486 + 2727.272728 Combine like terms: 2697.904605486 + 2727.272728 = 5425.177333486 x = 5425.177333486 Simplifying x = 5425.177333486Subproblem 2
x + -2727.272728 = -2697.904605486 Simplifying x + -2727.272728 = -2697.904605486 Reorder the terms: -2727.272728 + x = -2697.904605486 Solving -2727.272728 + x = -2697.904605486 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2727.272728' to each side of the equation. -2727.272728 + 2727.272728 + x = -2697.904605486 + 2727.272728 Combine like terms: -2727.272728 + 2727.272728 = 0.000000 0.000000 + x = -2697.904605486 + 2727.272728 x = -2697.904605486 + 2727.272728 Combine like terms: -2697.904605486 + 2727.272728 = 29.368122514 x = 29.368122514 Simplifying x = 29.368122514Solution
The solution to the problem is based on the solutions from the subproblems. x = {5425.177333486, 29.368122514}
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