3z^2+54z+243=1

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Solution for 3z^2+54z+243=1 equation:


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 = -8.422649734

Subproblem 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 = -9.577350266

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-8.422649734, -9.577350266}

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