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Simplifying 16000000 = 1 + x2 + -8000x + 16000000 Reorder the terms: 16000000 = 1 + 16000000 + -8000x + x2 Combine like terms: 1 + 16000000 = 16000001 16000000 = 16000001 + -8000x + x2 Solving 16000000 = 16000001 + -8000x + x2 Solving for variable 'x'. Combine like terms: 16000000 + -16000001 = -1 -1 + 8000x + -1x2 = 16000001 + -8000x + x2 + -16000001 + 8000x + -1x2 Reorder the terms: -1 + 8000x + -1x2 = 16000001 + -16000001 + -8000x + 8000x + x2 + -1x2 Combine like terms: 16000001 + -16000001 = 0 -1 + 8000x + -1x2 = 0 + -8000x + 8000x + x2 + -1x2 -1 + 8000x + -1x2 = -8000x + 8000x + x2 + -1x2 Combine like terms: -8000x + 8000x = 0 -1 + 8000x + -1x2 = 0 + x2 + -1x2 -1 + 8000x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -1 + 8000x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -8000x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -8000x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + -8000x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -8000x + x2 = 0 + -1 -8000x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 -8000x + x2 = -1 The x term is -8000x. Take half its coefficient (-4000). Square it (16000000) and add it to both sides. Add '16000000' to each side of the equation. -8000x + 16000000 + x2 = -1 + 16000000 Reorder the terms: 16000000 + -8000x + x2 = -1 + 16000000 Combine like terms: -1 + 16000000 = 15999999 16000000 + -8000x + x2 = 15999999 Factor a perfect square on the left side: (x + -4000)(x + -4000) = 15999999 Calculate the square root of the right side: 3999.999875 Break this problem into two subproblems by setting (x + -4000) equal to 3999.999875 and -3999.999875.Subproblem 1
x + -4000 = 3999.999875 Simplifying x + -4000 = 3999.999875 Reorder the terms: -4000 + x = 3999.999875 Solving -4000 + x = 3999.999875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4000' to each side of the equation. -4000 + 4000 + x = 3999.999875 + 4000 Combine like terms: -4000 + 4000 = 0 0 + x = 3999.999875 + 4000 x = 3999.999875 + 4000 Combine like terms: 3999.999875 + 4000 = 7999.999875 x = 7999.999875 Simplifying x = 7999.999875Subproblem 2
x + -4000 = -3999.999875 Simplifying x + -4000 = -3999.999875 Reorder the terms: -4000 + x = -3999.999875 Solving -4000 + x = -3999.999875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4000' to each side of the equation. -4000 + 4000 + x = -3999.999875 + 4000 Combine like terms: -4000 + 4000 = 0 0 + x = -3999.999875 + 4000 x = -3999.999875 + 4000 Combine like terms: -3999.999875 + 4000 = 0.000125 x = 0.000125 Simplifying x = 0.000125Solution
The solution to the problem is based on the solutions from the subproblems. x = {7999.999875, 0.000125}
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