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Simplifying x2 + x = 1000000 Reorder the terms: x + x2 = 1000000 Solving x + x2 = 1000000 Solving for variable 'x'. Reorder the terms: -1000000 + x + x2 = 1000000 + -1000000 Combine like terms: 1000000 + -1000000 = 0 -1000000 + x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '1000000' to each side of the equation. -1000000 + x + 1000000 + x2 = 0 + 1000000 Reorder the terms: -1000000 + 1000000 + x + x2 = 0 + 1000000 Combine like terms: -1000000 + 1000000 = 0 0 + x + x2 = 0 + 1000000 x + x2 = 0 + 1000000 Combine like terms: 0 + 1000000 = 1000000 x + x2 = 1000000 The x term is x. 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 = 1000000 + 0.25 Reorder the terms: 0.25 + x + x2 = 1000000 + 0.25 Combine like terms: 1000000 + 0.25 = 1000000.25 0.25 + x + x2 = 1000000.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 1000000.25 Calculate the square root of the right side: 1000.000125 Break this problem into two subproblems by setting (x + 0.5) equal to 1000.000125 and -1000.000125.Subproblem 1
x + 0.5 = 1000.000125 Simplifying x + 0.5 = 1000.000125 Reorder the terms: 0.5 + x = 1000.000125 Solving 0.5 + x = 1000.000125 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 = 1000.000125 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1000.000125 + -0.5 x = 1000.000125 + -0.5 Combine like terms: 1000.000125 + -0.5 = 999.500125 x = 999.500125 Simplifying x = 999.500125Subproblem 2
x + 0.5 = -1000.000125 Simplifying x + 0.5 = -1000.000125 Reorder the terms: 0.5 + x = -1000.000125 Solving 0.5 + x = -1000.000125 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 = -1000.000125 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1000.000125 + -0.5 x = -1000.000125 + -0.5 Combine like terms: -1000.000125 + -0.5 = -1000.500125 x = -1000.500125 Simplifying x = -1000.500125Solution
The solution to the problem is based on the solutions from the subproblems. x = {999.500125, -1000.500125}
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