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Simplifying -2z2 + -50z = 82 Reorder the terms: -50z + -2z2 = 82 Solving -50z + -2z2 = 82 Solving for variable 'z'. Reorder the terms: -82 + -50z + -2z2 = 82 + -82 Combine like terms: 82 + -82 = 0 -82 + -50z + -2z2 = 0 Factor out the Greatest Common Factor (GCF), '-2'. -2(41 + 25z + z2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(41 + 25z + z2)' equal to zero and attempt to solve: Simplifying 41 + 25z + z2 = 0 Solving 41 + 25z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-41' to each side of the equation. 41 + 25z + -41 + z2 = 0 + -41 Reorder the terms: 41 + -41 + 25z + z2 = 0 + -41 Combine like terms: 41 + -41 = 0 0 + 25z + z2 = 0 + -41 25z + z2 = 0 + -41 Combine like terms: 0 + -41 = -41 25z + z2 = -41 The z term is 25z. Take half its coefficient (12.5). Square it (156.25) and add it to both sides. Add '156.25' to each side of the equation. 25z + 156.25 + z2 = -41 + 156.25 Reorder the terms: 156.25 + 25z + z2 = -41 + 156.25 Combine like terms: -41 + 156.25 = 115.25 156.25 + 25z + z2 = 115.25 Factor a perfect square on the left side: (z + 12.5)(z + 12.5) = 115.25 Calculate the square root of the right side: 10.735455277 Break this problem into two subproblems by setting (z + 12.5) equal to 10.735455277 and -10.735455277.Subproblem 1
z + 12.5 = 10.735455277 Simplifying z + 12.5 = 10.735455277 Reorder the terms: 12.5 + z = 10.735455277 Solving 12.5 + z = 10.735455277 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + z = 10.735455277 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + z = 10.735455277 + -12.5 z = 10.735455277 + -12.5 Combine like terms: 10.735455277 + -12.5 = -1.764544723 z = -1.764544723 Simplifying z = -1.764544723Subproblem 2
z + 12.5 = -10.735455277 Simplifying z + 12.5 = -10.735455277 Reorder the terms: 12.5 + z = -10.735455277 Solving 12.5 + z = -10.735455277 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + z = -10.735455277 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + z = -10.735455277 + -12.5 z = -10.735455277 + -12.5 Combine like terms: -10.735455277 + -12.5 = -23.235455277 z = -23.235455277 Simplifying z = -23.235455277Solution
The solution to the problem is based on the solutions from the subproblems. z = {-1.764544723, -23.235455277}Solution
z = {-1.764544723, -23.235455277}
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