3z^2=9z+27

Solution for 3z^2=9z+27 equation:

Simplifying
3z2 = 9z + 27

Reorder the terms:
3z2 = 27 + 9z

Solving
3z2 = 27 + 9z

Solving for variable 'z'.

Reorder the terms:
-27 + -9z + 3z2 = 27 + 9z + -27 + -9z

Reorder the terms:
-27 + -9z + 3z2 = 27 + -27 + 9z + -9z

Combine like terms: 27 + -27 = 0
-27 + -9z + 3z2 = 0 + 9z + -9z
-27 + -9z + 3z2 = 9z + -9z

Combine like terms: 9z + -9z = 0
-27 + -9z + 3z2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-9 + -3z + z2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-9 + -3z + z2)' equal to zero and attempt to solve:

Simplifying
-9 + -3z + z2 = 0

Solving
-9 + -3z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -3z + 9 + z2 = 0 + 9

Reorder the terms:
-9 + 9 + -3z + z2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -3z + z2 = 0 + 9
-3z + z2 = 0 + 9

Combine like terms: 0 + 9 = 9
-3z + z2 = 9

The z term is -3z.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3z + 2.25 + z2 = 9 + 2.25

Reorder the terms:
2.25 + -3z + z2 = 9 + 2.25

Combine like terms: 9 + 2.25 = 11.25
2.25 + -3z + z2 = 11.25

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(z + -1.5) equal to 3.354101966 and -3.354101966.

Subproblem 1
z + -1.5 = 3.354101966

Simplifying
z + -1.5 = 3.354101966

Reorder the terms:
-1.5 + z = 3.354101966

Solving
-1.5 + z = 3.354101966

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '1.5' to each side of the equation.
-1.5 + 1.5 + z = 3.354101966 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = 3.354101966 + 1.5
z = 3.354101966 + 1.5

Combine like terms: 3.354101966 + 1.5 = 4.854101966
z = 4.854101966

Simplifying
z = 4.854101966

Subproblem 2
z + -1.5 = -3.354101966

Simplifying
z + -1.5 = -3.354101966

Reorder the terms:
-1.5 + z = -3.354101966

Solving
-1.5 + z = -3.354101966

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '1.5' to each side of the equation.
-1.5 + 1.5 + z = -3.354101966 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = -3.354101966 + 1.5
z = -3.354101966 + 1.5

Combine like terms: -3.354101966 + 1.5 = -1.854101966
z = -1.854101966

Simplifying
z = -1.854101966

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.854101966, -1.854101966}
Solution
z = {4.854101966, -1.854101966}