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Simplifying 350x + -0.025x2 = 200000 + 130x + -0.005x2 Solving 350x + -0.025x2 = 200000 + 130x + -0.005x2 Solving for variable 'x'. Reorder the terms: -200000 + 350x + -130x + -0.025x2 + 0.005x2 = 200000 + 130x + -0.005x2 + -200000 + -130x + 0.005x2 Combine like terms: 350x + -130x = 220x -200000 + 220x + -0.025x2 + 0.005x2 = 200000 + 130x + -0.005x2 + -200000 + -130x + 0.005x2 Combine like terms: -0.025x2 + 0.005x2 = -0.02x2 -200000 + 220x + -0.02x2 = 200000 + 130x + -0.005x2 + -200000 + -130x + 0.005x2 Reorder the terms: -200000 + 220x + -0.02x2 = 200000 + -200000 + 130x + -130x + -0.005x2 + 0.005x2 Combine like terms: 200000 + -200000 = 0 -200000 + 220x + -0.02x2 = 0 + 130x + -130x + -0.005x2 + 0.005x2 -200000 + 220x + -0.02x2 = 130x + -130x + -0.005x2 + 0.005x2 Combine like terms: 130x + -130x = 0 -200000 + 220x + -0.02x2 = 0 + -0.005x2 + 0.005x2 -200000 + 220x + -0.02x2 = -0.005x2 + 0.005x2 Combine like terms: -0.005x2 + 0.005x2 = 0.000 -200000 + 220x + -0.02x2 = 0.000 Begin completing the square. Divide all terms by -0.02 the coefficient of the squared term: Divide each side by '-0.02'. 10000000 + -11000x + x2 = 0.0 Move the constant term to the right: Add '-10000000' to each side of the equation. 10000000 + -11000x + -10000000 + x2 = 0.0 + -10000000 Reorder the terms: 10000000 + -10000000 + -11000x + x2 = 0.0 + -10000000 Combine like terms: 10000000 + -10000000 = 0 0 + -11000x + x2 = 0.0 + -10000000 -11000x + x2 = 0.0 + -10000000 Combine like terms: 0.0 + -10000000 = -10000000 -11000x + x2 = -10000000 The x term is -11000x. Take half its coefficient (-5500). Square it (30250000) and add it to both sides. Add '30250000' to each side of the equation. -11000x + 30250000 + x2 = -10000000 + 30250000 Reorder the terms: 30250000 + -11000x + x2 = -10000000 + 30250000 Combine like terms: -10000000 + 30250000 = 20250000 30250000 + -11000x + x2 = 20250000 Factor a perfect square on the left side: (x + -5500)(x + -5500) = 20250000 Calculate the square root of the right side: 4500 Break this problem into two subproblems by setting (x + -5500) equal to 4500 and -4500.Subproblem 1
x + -5500 = 4500 Simplifying x + -5500 = 4500 Reorder the terms: -5500 + x = 4500 Solving -5500 + x = 4500 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5500' to each side of the equation. -5500 + 5500 + x = 4500 + 5500 Combine like terms: -5500 + 5500 = 0 0 + x = 4500 + 5500 x = 4500 + 5500 Combine like terms: 4500 + 5500 = 10000 x = 10000 Simplifying x = 10000Subproblem 2
x + -5500 = -4500 Simplifying x + -5500 = -4500 Reorder the terms: -5500 + x = -4500 Solving -5500 + x = -4500 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5500' to each side of the equation. -5500 + 5500 + x = -4500 + 5500 Combine like terms: -5500 + 5500 = 0 0 + x = -4500 + 5500 x = -4500 + 5500 Combine like terms: -4500 + 5500 = 1000 x = 1000 Simplifying x = 1000Solution
The solution to the problem is based on the solutions from the subproblems. x = {10000, 1000}
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