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Simplifying (z) * 2443.73(1 + -0.17 + -1z) * 584.28 * 0.17 * 908.79 = 561.47 Combine like terms: 1 + -0.17 = 0.83 z * 2443.73(0.83 + -1z) * 584.28 * 0.17 * 908.79 = 561.47 Reorder the terms for easier multiplication: 2443.73 * 584.28 * 0.17 * 908.79z(0.83 + -1z) = 561.47 Multiply 2443.73 * 584.28 1427822.564 * 0.17 * 908.79z(0.83 + -1z) = 561.47 Multiply 1427822.564 * 0.17 242729.8359 * 908.79z(0.83 + -1z) = 561.47 Multiply 242729.8359 * 908.79 220590447.6z(0.83 + -1z) = 561.47 (0.83 * 220590447.6z + -1z * 220590447.6z) = 561.47 (183090071.5z + -220590447.6z2) = 561.47 Solving 183090071.5z + -220590447.6z2 = 561.47 Solving for variable 'z'. Reorder the terms: -561.47 + 183090071.5z + -220590447.6z2 = 561.47 + -561.47 Combine like terms: 561.47 + -561.47 = 0.00 -561.47 + 183090071.5z + -220590447.6z2 = 0.00 Begin completing the square. Divide all terms by -220590447.6 the coefficient of the squared term: Divide each side by '-220590447.6'. 0.000002545305139 + -0.83z + z2 = 0.0 Move the constant term to the right: Add '-0.000002545305139' to each side of the equation. 0.000002545305139 + -0.83z + -0.000002545305139 + z2 = 0.0 + -0.000002545305139 Reorder the terms: 0.000002545305139 + -0.000002545305139 + -0.83z + z2 = 0.0 + -0.000002545305139 Combine like terms: 0.000002545305139 + -0.000002545305139 = 0.000000000000000 0.000000000000000 + -0.83z + z2 = 0.0 + -0.000002545305139 -0.83z + z2 = 0.0 + -0.000002545305139 Combine like terms: 0.0 + -0.000002545305139 = -0.000002545305139 -0.83z + z2 = -0.000002545305139 The z term is -0.83z. 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. -0.83z + 0.172225 + z2 = -0.000002545305139 + 0.172225 Reorder the terms: 0.172225 + -0.83z + z2 = -0.000002545305139 + 0.172225 Combine like terms: -0.000002545305139 + 0.172225 = 0.172222454694861 0.172225 + -0.83z + z2 = 0.172222454694861 Factor a perfect square on the left side: (z + -0.415)(z + -0.415) = 0.172222454694861 Calculate the square root of the right side: 0.414996933 Break this problem into two subproblems by setting (z + -0.415) equal to 0.414996933 and -0.414996933.Subproblem 1
z + -0.415 = 0.414996933 Simplifying z + -0.415 = 0.414996933 Reorder the terms: -0.415 + z = 0.414996933 Solving -0.415 + z = 0.414996933 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 = 0.414996933 + 0.415 Combine like terms: -0.415 + 0.415 = 0.000 0.000 + z = 0.414996933 + 0.415 z = 0.414996933 + 0.415 Combine like terms: 0.414996933 + 0.415 = 0.829996933 z = 0.829996933 Simplifying z = 0.829996933Subproblem 2
z + -0.415 = -0.414996933 Simplifying z + -0.415 = -0.414996933 Reorder the terms: -0.415 + z = -0.414996933 Solving -0.415 + z = -0.414996933 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 = -0.414996933 + 0.415 Combine like terms: -0.415 + 0.415 = 0.000 0.000 + z = -0.414996933 + 0.415 z = -0.414996933 + 0.415 Combine like terms: -0.414996933 + 0.415 = 0.000003067 z = 0.000003067 Simplifying z = 0.000003067Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.829996933, 0.000003067}
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