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Simplifying 46z2 + 96z + 15 = 0 Reorder the terms: 15 + 96z + 46z2 = 0 Solving 15 + 96z + 46z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 46 the coefficient of the squared term: Divide each side by '46'. 0.3260869565 + 2.086956522z + z2 = 0 Move the constant term to the right: Add '-0.3260869565' to each side of the equation. 0.3260869565 + 2.086956522z + -0.3260869565 + z2 = 0 + -0.3260869565 Reorder the terms: 0.3260869565 + -0.3260869565 + 2.086956522z + z2 = 0 + -0.3260869565 Combine like terms: 0.3260869565 + -0.3260869565 = 0.0000000000 0.0000000000 + 2.086956522z + z2 = 0 + -0.3260869565 2.086956522z + z2 = 0 + -0.3260869565 Combine like terms: 0 + -0.3260869565 = -0.3260869565 2.086956522z + z2 = -0.3260869565 The z term is 2.086956522z. Take half its coefficient (1.043478261). Square it (1.088846881) and add it to both sides. Add '1.088846881' to each side of the equation. 2.086956522z + 1.088846881 + z2 = -0.3260869565 + 1.088846881 Reorder the terms: 1.088846881 + 2.086956522z + z2 = -0.3260869565 + 1.088846881 Combine like terms: -0.3260869565 + 1.088846881 = 0.7627599245 1.088846881 + 2.086956522z + z2 = 0.7627599245 Factor a perfect square on the left side: (z + 1.043478261)(z + 1.043478261) = 0.7627599245 Calculate the square root of the right side: 0.873361279 Break this problem into two subproblems by setting (z + 1.043478261) equal to 0.873361279 and -0.873361279.Subproblem 1
z + 1.043478261 = 0.873361279 Simplifying z + 1.043478261 = 0.873361279 Reorder the terms: 1.043478261 + z = 0.873361279 Solving 1.043478261 + z = 0.873361279 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.043478261' to each side of the equation. 1.043478261 + -1.043478261 + z = 0.873361279 + -1.043478261 Combine like terms: 1.043478261 + -1.043478261 = 0.000000000 0.000000000 + z = 0.873361279 + -1.043478261 z = 0.873361279 + -1.043478261 Combine like terms: 0.873361279 + -1.043478261 = -0.170116982 z = -0.170116982 Simplifying z = -0.170116982Subproblem 2
z + 1.043478261 = -0.873361279 Simplifying z + 1.043478261 = -0.873361279 Reorder the terms: 1.043478261 + z = -0.873361279 Solving 1.043478261 + z = -0.873361279 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.043478261' to each side of the equation. 1.043478261 + -1.043478261 + z = -0.873361279 + -1.043478261 Combine like terms: 1.043478261 + -1.043478261 = 0.000000000 0.000000000 + z = -0.873361279 + -1.043478261 z = -0.873361279 + -1.043478261 Combine like terms: -0.873361279 + -1.043478261 = -1.91683954 z = -1.91683954 Simplifying z = -1.91683954Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.170116982, -1.91683954}
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