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Simplifying x(x + 1) = 160000 Reorder the terms: x(1 + x) = 160000 (1 * x + x * x) = 160000 (1x + x2) = 160000 Solving 1x + x2 = 160000 Solving for variable 'x'. Reorder the terms: -160000 + 1x + x2 = 160000 + -160000 Combine like terms: 160000 + -160000 = 0 -160000 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '160000' to each side of the equation. -160000 + 1x + 160000 + x2 = 0 + 160000 Reorder the terms: -160000 + 160000 + 1x + x2 = 0 + 160000 Combine like terms: -160000 + 160000 = 0 0 + 1x + x2 = 0 + 160000 1x + x2 = 0 + 160000 Combine like terms: 0 + 160000 = 160000 1x + x2 = 160000 The x term is 1x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. x + 0.25 + x2 = 160000 + 0.25 Reorder the terms: 0.25 + x + x2 = 160000 + 0.25 Combine like terms: 160000 + 0.25 = 160000.25 0.25 + x + x2 = 160000.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 160000.25 Calculate the square root of the right side: 400.0003125 Break this problem into two subproblems by setting (x + 0.5) equal to 400.0003125 and -400.0003125.Subproblem 1
x + 0.5 = 400.0003125 Simplifying x + 0.5 = 400.0003125 Reorder the terms: 0.5 + x = 400.0003125 Solving 0.5 + x = 400.0003125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 400.0003125 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 400.0003125 + -0.5 x = 400.0003125 + -0.5 Combine like terms: 400.0003125 + -0.5 = 399.5003125 x = 399.5003125 Simplifying x = 399.5003125Subproblem 2
x + 0.5 = -400.0003125 Simplifying x + 0.5 = -400.0003125 Reorder the terms: 0.5 + x = -400.0003125 Solving 0.5 + x = -400.0003125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -400.0003125 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -400.0003125 + -0.5 x = -400.0003125 + -0.5 Combine like terms: -400.0003125 + -0.5 = -400.5003125 x = -400.5003125 Simplifying x = -400.5003125Solution
The solution to the problem is based on the solutions from the subproblems. x = {399.5003125, -400.5003125}
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