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Simplifying x * x + -1000x + -1000x + -1000 = 0 Multiply x * x x2 + -1000x + -1000x + -1000 = 0 Reorder the terms: -1000 + -1000x + -1000x + x2 = 0 Combine like terms: -1000x + -1000x = -2000x -1000 + -2000x + x2 = 0 Solving -1000 + -2000x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '1000' to each side of the equation. -1000 + -2000x + 1000 + x2 = 0 + 1000 Reorder the terms: -1000 + 1000 + -2000x + x2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + -2000x + x2 = 0 + 1000 -2000x + x2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 -2000x + x2 = 1000 The x term is -2000x. Take half its coefficient (-1000). Square it (1000000) and add it to both sides. Add '1000000' to each side of the equation. -2000x + 1000000 + x2 = 1000 + 1000000 Reorder the terms: 1000000 + -2000x + x2 = 1000 + 1000000 Combine like terms: 1000 + 1000000 = 1001000 1000000 + -2000x + x2 = 1001000 Factor a perfect square on the left side: (x + -1000)(x + -1000) = 1001000 Calculate the square root of the right side: 1000.499875062 Break this problem into two subproblems by setting (x + -1000) equal to 1000.499875062 and -1000.499875062.Subproblem 1
x + -1000 = 1000.499875062 Simplifying x + -1000 = 1000.499875062 Reorder the terms: -1000 + x = 1000.499875062 Solving -1000 + x = 1000.499875062 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1000' to each side of the equation. -1000 + 1000 + x = 1000.499875062 + 1000 Combine like terms: -1000 + 1000 = 0 0 + x = 1000.499875062 + 1000 x = 1000.499875062 + 1000 Combine like terms: 1000.499875062 + 1000 = 2000.499875062 x = 2000.499875062 Simplifying x = 2000.499875062Subproblem 2
x + -1000 = -1000.499875062 Simplifying x + -1000 = -1000.499875062 Reorder the terms: -1000 + x = -1000.499875062 Solving -1000 + x = -1000.499875062 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1000' to each side of the equation. -1000 + 1000 + x = -1000.499875062 + 1000 Combine like terms: -1000 + 1000 = 0 0 + x = -1000.499875062 + 1000 x = -1000.499875062 + 1000 Combine like terms: -1000.499875062 + 1000 = -0.499875062 x = -0.499875062 Simplifying x = -0.499875062Solution
The solution to the problem is based on the solutions from the subproblems. x = {2000.499875062, -0.499875062}
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