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Simplifying z2 + -1000z + 10100 = 0 Reorder the terms: 10100 + -1000z + z2 = 0 Solving 10100 + -1000z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-10100' to each side of the equation. 10100 + -1000z + -10100 + z2 = 0 + -10100 Reorder the terms: 10100 + -10100 + -1000z + z2 = 0 + -10100 Combine like terms: 10100 + -10100 = 0 0 + -1000z + z2 = 0 + -10100 -1000z + z2 = 0 + -10100 Combine like terms: 0 + -10100 = -10100 -1000z + z2 = -10100 The z term is -1000z. Take half its coefficient (-500). Square it (250000) and add it to both sides. Add '250000' to each side of the equation. -1000z + 250000 + z2 = -10100 + 250000 Reorder the terms: 250000 + -1000z + z2 = -10100 + 250000 Combine like terms: -10100 + 250000 = 239900 250000 + -1000z + z2 = 239900 Factor a perfect square on the left side: (z + -500)(z + -500) = 239900 Calculate the square root of the right side: 489.79587585 Break this problem into two subproblems by setting (z + -500) equal to 489.79587585 and -489.79587585.Subproblem 1
z + -500 = 489.79587585 Simplifying z + -500 = 489.79587585 Reorder the terms: -500 + z = 489.79587585 Solving -500 + z = 489.79587585 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + z = 489.79587585 + 500 Combine like terms: -500 + 500 = 0 0 + z = 489.79587585 + 500 z = 489.79587585 + 500 Combine like terms: 489.79587585 + 500 = 989.79587585 z = 989.79587585 Simplifying z = 989.79587585Subproblem 2
z + -500 = -489.79587585 Simplifying z + -500 = -489.79587585 Reorder the terms: -500 + z = -489.79587585 Solving -500 + z = -489.79587585 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + z = -489.79587585 + 500 Combine like terms: -500 + 500 = 0 0 + z = -489.79587585 + 500 z = -489.79587585 + 500 Combine like terms: -489.79587585 + 500 = 10.20412415 z = 10.20412415 Simplifying z = 10.20412415Solution
The solution to the problem is based on the solutions from the subproblems. z = {989.79587585, 10.20412415}
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