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Simplifying 1000z + -1z2 = 10 Solving 1000z + -1z2 = 10 Solving for variable 'z'. Reorder the terms: -10 + 1000z + -1z2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + 1000z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 10 + -1000z + z2 = 0 Move the constant term to the right: Add '-10' to each side of the equation. 10 + -1000z + -10 + z2 = 0 + -10 Reorder the terms: 10 + -10 + -1000z + z2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + -1000z + z2 = 0 + -10 -1000z + z2 = 0 + -10 Combine like terms: 0 + -10 = -10 -1000z + z2 = -10 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 = -10 + 250000 Reorder the terms: 250000 + -1000z + z2 = -10 + 250000 Combine like terms: -10 + 250000 = 249990 250000 + -1000z + z2 = 249990 Factor a perfect square on the left side: (z + -500)(z + -500) = 249990 Calculate the square root of the right side: 499.9899999 Break this problem into two subproblems by setting (z + -500) equal to 499.9899999 and -499.9899999.Subproblem 1
z + -500 = 499.9899999 Simplifying z + -500 = 499.9899999 Reorder the terms: -500 + z = 499.9899999 Solving -500 + z = 499.9899999 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 = 499.9899999 + 500 Combine like terms: -500 + 500 = 0 0 + z = 499.9899999 + 500 z = 499.9899999 + 500 Combine like terms: 499.9899999 + 500 = 999.9899999 z = 999.9899999 Simplifying z = 999.9899999Subproblem 2
z + -500 = -499.9899999 Simplifying z + -500 = -499.9899999 Reorder the terms: -500 + z = -499.9899999 Solving -500 + z = -499.9899999 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 = -499.9899999 + 500 Combine like terms: -500 + 500 = 0 0 + z = -499.9899999 + 500 z = -499.9899999 + 500 Combine like terms: -499.9899999 + 500 = 0.0100001 z = 0.0100001 Simplifying z = 0.0100001Solution
The solution to the problem is based on the solutions from the subproblems. z = {999.9899999, 0.0100001}
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