36z^2-51z+18=z

Solution for 36z^2-51z+18=z equation:

Simplifying
36z2 + -51z + 18 = z

Reorder the terms:
18 + -51z + 36z2 = z

Solving
18 + -51z + 36z2 = z

Solving for variable 'z'.

Reorder the terms:
18 + -51z + -1z + 36z2 = z + -1z

Combine like terms: -51z + -1z = -52z
18 + -52z + 36z2 = z + -1z

Combine like terms: z + -1z = 0
18 + -52z + 36z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(9 + -26z + 18z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(9 + -26z + 18z2)' equal to zero and attempt to solve:

Simplifying
9 + -26z + 18z2 = 0

Solving
9 + -26z + 18z2 = 0

Begin completing the square.  Divide all terms by
18 the coefficient of the squared term: 

Divide each side by '18'.
0.5 + -1.444444444z + z2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -1.444444444z + -0.5 + z2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -1.444444444z + z2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -1.444444444z + z2 = 0 + -0.5
-1.444444444z + z2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-1.444444444z + z2 = -0.5

The z term is -1.444444444z.  Take half its coefficient (-0.722222222).
Square it (0.5216049380) and add it to both sides.

Add '0.5216049380' to each side of the equation.
-1.444444444z + 0.5216049380 + z2 = -0.5 + 0.5216049380

Reorder the terms:
0.5216049380 + -1.444444444z + z2 = -0.5 + 0.5216049380

Combine like terms: -0.5 + 0.5216049380 = 0.021604938
0.5216049380 + -1.444444444z + z2 = 0.021604938

Factor a perfect square on the left side:
(z + -0.722222222)(z + -0.722222222) = 0.021604938

Calculate the square root of the right side: 0.146986183

Break this problem into two subproblems by setting 
(z + -0.722222222) equal to 0.146986183 and -0.146986183.

Subproblem 1
z + -0.722222222 = 0.146986183

Simplifying
z + -0.722222222 = 0.146986183

Reorder the terms:
-0.722222222 + z = 0.146986183

Solving
-0.722222222 + z = 0.146986183

Solving for variable 'z'.

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

Add '0.722222222' to each side of the equation.
-0.722222222 + 0.722222222 + z = 0.146986183 + 0.722222222

Combine like terms: -0.722222222 + 0.722222222 = 0.000000000
0.000000000 + z = 0.146986183 + 0.722222222
z = 0.146986183 + 0.722222222

Combine like terms: 0.146986183 + 0.722222222 = 0.869208405
z = 0.869208405

Simplifying
z = 0.869208405

Subproblem 2
z + -0.722222222 = -0.146986183

Simplifying
z + -0.722222222 = -0.146986183

Reorder the terms:
-0.722222222 + z = -0.146986183

Solving
-0.722222222 + z = -0.146986183

Solving for variable 'z'.

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

Add '0.722222222' to each side of the equation.
-0.722222222 + 0.722222222 + z = -0.146986183 + 0.722222222

Combine like terms: -0.722222222 + 0.722222222 = 0.000000000
0.000000000 + z = -0.146986183 + 0.722222222
z = -0.146986183 + 0.722222222

Combine like terms: -0.146986183 + 0.722222222 = 0.575236039
z = 0.575236039

Simplifying
z = 0.575236039

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.869208405, 0.575236039}
Solution
z = {0.869208405, 0.575236039}