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Simplifying z(2443.73)(1 + -0.17 + -1z) * 584.28 + 154.49 + -561.47 = 0 Combine like terms: 1 + -0.17 = 0.83 z * 2443.73(0.83 + -1z) * 584.28 + 154.49 + -561.47 = 0 Reorder the terms for easier multiplication: 2443.73 * 584.28z(0.83 + -1z) + 154.49 + -561.47 = 0 Multiply 2443.73 * 584.28 1427822.564z(0.83 + -1z) + 154.49 + -561.47 = 0 (0.83 * 1427822.564z + -1z * 1427822.564z) + 154.49 + -561.47 = 0 (1185092.728z + -1427822.564z2) + 154.49 + -561.47 = 0 Reorder the terms: 154.49 + -561.47 + 1185092.728z + -1427822.564z2 = 0 Combine like terms: 154.49 + -561.47 = -406.98 -406.98 + 1185092.728z + -1427822.564z2 = 0 Solving -406.98 + 1185092.728z + -1427822.564z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -1427822.564 the coefficient of the squared term: Divide each side by '-1427822.564'. 0.000285035417 + -0.8299999999z + z2 = 0 Move the constant term to the right: Add '-0.000285035417' to each side of the equation. 0.000285035417 + -0.8299999999z + -0.000285035417 + z2 = 0 + -0.000285035417 Reorder the terms: 0.000285035417 + -0.000285035417 + -0.8299999999z + z2 = 0 + -0.000285035417 Combine like terms: 0.000285035417 + -0.000285035417 = 0.000000000000 0.000000000000 + -0.8299999999z + z2 = 0 + -0.000285035417 -0.8299999999z + z2 = 0 + -0.000285035417 Combine like terms: 0 + -0.000285035417 = -0.000285035417 -0.8299999999z + z2 = -0.000285035417 The z term is -0.8299999999z. Take half its coefficient (-0.415). Square it (0.172225) and add it to both sides. Add '0.172225' to each side of the equation. -0.8299999999z + 0.172225 + z2 = -0.000285035417 + 0.172225 Reorder the terms: 0.172225 + -0.8299999999z + z2 = -0.000285035417 + 0.172225 Combine like terms: -0.000285035417 + 0.172225 = 0.171939964583 0.172225 + -0.8299999999z + z2 = 0.171939964583 Factor a perfect square on the left side: (z + -0.415)(z + -0.415) = 0.171939964583 Calculate the square root of the right side: 0.414656442 Break this problem into two subproblems by setting (z + -0.415) equal to 0.414656442 and -0.414656442.Subproblem 1
z + -0.415 = 0.414656442 Simplifying z + -0.415 = 0.414656442 Reorder the terms: -0.415 + z = 0.414656442 Solving -0.415 + z = 0.414656442 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.415' to each side of the equation. -0.415 + 0.415 + z = 0.414656442 + 0.415 Combine like terms: -0.415 + 0.415 = 0.000 0.000 + z = 0.414656442 + 0.415 z = 0.414656442 + 0.415 Combine like terms: 0.414656442 + 0.415 = 0.829656442 z = 0.829656442 Simplifying z = 0.829656442Subproblem 2
z + -0.415 = -0.414656442 Simplifying z + -0.415 = -0.414656442 Reorder the terms: -0.415 + z = -0.414656442 Solving -0.415 + z = -0.414656442 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.415' to each side of the equation. -0.415 + 0.415 + z = -0.414656442 + 0.415 Combine like terms: -0.415 + 0.415 = 0.000 0.000 + z = -0.414656442 + 0.415 z = -0.414656442 + 0.415 Combine like terms: -0.414656442 + 0.415 = 0.000343558 z = 0.000343558 Simplifying z = 0.000343558Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.829656442, 0.000343558}
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