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Simplifying z2 = -45 + -15z Solving z2 = -45 + -15z Solving for variable 'z'. Reorder the terms: 45 + 15z + z2 = -45 + -15z + 45 + 15z Reorder the terms: 45 + 15z + z2 = -45 + 45 + -15z + 15z Combine like terms: -45 + 45 = 0 45 + 15z + z2 = 0 + -15z + 15z 45 + 15z + z2 = -15z + 15z Combine like terms: -15z + 15z = 0 45 + 15z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-45' to each side of the equation. 45 + 15z + -45 + z2 = 0 + -45 Reorder the terms: 45 + -45 + 15z + z2 = 0 + -45 Combine like terms: 45 + -45 = 0 0 + 15z + z2 = 0 + -45 15z + z2 = 0 + -45 Combine like terms: 0 + -45 = -45 15z + z2 = -45 The z term is 15z. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15z + 56.25 + z2 = -45 + 56.25 Reorder the terms: 56.25 + 15z + z2 = -45 + 56.25 Combine like terms: -45 + 56.25 = 11.25 56.25 + 15z + z2 = 11.25 Factor a perfect square on the left side: (z + 7.5)(z + 7.5) = 11.25 Calculate the square root of the right side: 3.354101966 Break this problem into two subproblems by setting (z + 7.5) equal to 3.354101966 and -3.354101966.Subproblem 1
z + 7.5 = 3.354101966 Simplifying z + 7.5 = 3.354101966 Reorder the terms: 7.5 + z = 3.354101966 Solving 7.5 + z = 3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + z = 3.354101966 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + z = 3.354101966 + -7.5 z = 3.354101966 + -7.5 Combine like terms: 3.354101966 + -7.5 = -4.145898034 z = -4.145898034 Simplifying z = -4.145898034Subproblem 2
z + 7.5 = -3.354101966 Simplifying z + 7.5 = -3.354101966 Reorder the terms: 7.5 + z = -3.354101966 Solving 7.5 + z = -3.354101966 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + z = -3.354101966 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + z = -3.354101966 + -7.5 z = -3.354101966 + -7.5 Combine like terms: -3.354101966 + -7.5 = -10.854101966 z = -10.854101966 Simplifying z = -10.854101966Solution
The solution to the problem is based on the solutions from the subproblems. z = {-4.145898034, -10.854101966}
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