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Simplifying 8z2 + -9z + -82 = 0 Reorder the terms: -82 + -9z + 8z2 = 0 Solving -82 + -9z + 8z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -10.25 + -1.125z + z2 = 0 Move the constant term to the right: Add '10.25' to each side of the equation. -10.25 + -1.125z + 10.25 + z2 = 0 + 10.25 Reorder the terms: -10.25 + 10.25 + -1.125z + z2 = 0 + 10.25 Combine like terms: -10.25 + 10.25 = 0.00 0.00 + -1.125z + z2 = 0 + 10.25 -1.125z + z2 = 0 + 10.25 Combine like terms: 0 + 10.25 = 10.25 -1.125z + z2 = 10.25 The z term is -1.125z. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125z + 0.31640625 + z2 = 10.25 + 0.31640625 Reorder the terms: 0.31640625 + -1.125z + z2 = 10.25 + 0.31640625 Combine like terms: 10.25 + 0.31640625 = 10.56640625 0.31640625 + -1.125z + z2 = 10.56640625 Factor a perfect square on the left side: (z + -0.5625)(z + -0.5625) = 10.56640625 Calculate the square root of the right side: 3.250600906 Break this problem into two subproblems by setting (z + -0.5625) equal to 3.250600906 and -3.250600906.Subproblem 1
z + -0.5625 = 3.250600906 Simplifying z + -0.5625 = 3.250600906 Reorder the terms: -0.5625 + z = 3.250600906 Solving -0.5625 + z = 3.250600906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + z = 3.250600906 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + z = 3.250600906 + 0.5625 z = 3.250600906 + 0.5625 Combine like terms: 3.250600906 + 0.5625 = 3.813100906 z = 3.813100906 Simplifying z = 3.813100906Subproblem 2
z + -0.5625 = -3.250600906 Simplifying z + -0.5625 = -3.250600906 Reorder the terms: -0.5625 + z = -3.250600906 Solving -0.5625 + z = -3.250600906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + z = -3.250600906 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + z = -3.250600906 + 0.5625 z = -3.250600906 + 0.5625 Combine like terms: -3.250600906 + 0.5625 = -2.688100906 z = -2.688100906 Simplifying z = -2.688100906Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.813100906, -2.688100906}
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