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Simplifying 3z2 + 54z + 243 = 1 Reorder the terms: 243 + 54z + 3z2 = 1 Solving 243 + 54z + 3z2 = 1 Solving for variable 'z'. Reorder the terms: 243 + -1 + 54z + 3z2 = 1 + -1 Combine like terms: 243 + -1 = 242 242 + 54z + 3z2 = 1 + -1 Combine like terms: 1 + -1 = 0 242 + 54z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 80.66666667 + 18z + z2 = 0 Move the constant term to the right: Add '-80.66666667' to each side of the equation. 80.66666667 + 18z + -80.66666667 + z2 = 0 + -80.66666667 Reorder the terms: 80.66666667 + -80.66666667 + 18z + z2 = 0 + -80.66666667 Combine like terms: 80.66666667 + -80.66666667 = 0.00000000 0.00000000 + 18z + z2 = 0 + -80.66666667 18z + z2 = 0 + -80.66666667 Combine like terms: 0 + -80.66666667 = -80.66666667 18z + z2 = -80.66666667 The z term is 18z. Take half its coefficient (9). Square it (81) and add it to both sides. Add '81' to each side of the equation. 18z + 81 + z2 = -80.66666667 + 81 Reorder the terms: 81 + 18z + z2 = -80.66666667 + 81 Combine like terms: -80.66666667 + 81 = 0.33333333 81 + 18z + z2 = 0.33333333 Factor a perfect square on the left side: (z + 9)(z + 9) = 0.33333333 Calculate the square root of the right side: 0.577350266 Break this problem into two subproblems by setting (z + 9) equal to 0.577350266 and -0.577350266.Subproblem 1
z + 9 = 0.577350266 Simplifying z + 9 = 0.577350266 Reorder the terms: 9 + z = 0.577350266 Solving 9 + z = 0.577350266 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + z = 0.577350266 + -9 Combine like terms: 9 + -9 = 0 0 + z = 0.577350266 + -9 z = 0.577350266 + -9 Combine like terms: 0.577350266 + -9 = -8.422649734 z = -8.422649734 Simplifying z = -8.422649734Subproblem 2
z + 9 = -0.577350266 Simplifying z + 9 = -0.577350266 Reorder the terms: 9 + z = -0.577350266 Solving 9 + z = -0.577350266 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + z = -0.577350266 + -9 Combine like terms: 9 + -9 = 0 0 + z = -0.577350266 + -9 z = -0.577350266 + -9 Combine like terms: -0.577350266 + -9 = -9.577350266 z = -9.577350266 Simplifying z = -9.577350266Solution
The solution to the problem is based on the solutions from the subproblems. z = {-8.422649734, -9.577350266}
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