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Simplifying 3z2 + 27z + -44 = 0 Reorder the terms: -44 + 27z + 3z2 = 0 Solving -44 + 27z + 3z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -14.66666667 + 9z + z2 = 0 Move the constant term to the right: Add '14.66666667' to each side of the equation. -14.66666667 + 9z + 14.66666667 + z2 = 0 + 14.66666667 Reorder the terms: -14.66666667 + 14.66666667 + 9z + z2 = 0 + 14.66666667 Combine like terms: -14.66666667 + 14.66666667 = 0.00000000 0.00000000 + 9z + z2 = 0 + 14.66666667 9z + z2 = 0 + 14.66666667 Combine like terms: 0 + 14.66666667 = 14.66666667 9z + z2 = 14.66666667 The z term is 9z. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9z + 20.25 + z2 = 14.66666667 + 20.25 Reorder the terms: 20.25 + 9z + z2 = 14.66666667 + 20.25 Combine like terms: 14.66666667 + 20.25 = 34.91666667 20.25 + 9z + z2 = 34.91666667 Factor a perfect square on the left side: (z + 4.5)(z + 4.5) = 34.91666667 Calculate the square root of the right side: 5.909032634 Break this problem into two subproblems by setting (z + 4.5) equal to 5.909032634 and -5.909032634.Subproblem 1
z + 4.5 = 5.909032634 Simplifying z + 4.5 = 5.909032634 Reorder the terms: 4.5 + z = 5.909032634 Solving 4.5 + z = 5.909032634 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = 5.909032634 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 5.909032634 + -4.5 z = 5.909032634 + -4.5 Combine like terms: 5.909032634 + -4.5 = 1.409032634 z = 1.409032634 Simplifying z = 1.409032634Subproblem 2
z + 4.5 = -5.909032634 Simplifying z + 4.5 = -5.909032634 Reorder the terms: 4.5 + z = -5.909032634 Solving 4.5 + z = -5.909032634 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = -5.909032634 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -5.909032634 + -4.5 z = -5.909032634 + -4.5 Combine like terms: -5.909032634 + -4.5 = -10.409032634 z = -10.409032634 Simplifying z = -10.409032634Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.409032634, -10.409032634}
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