z^2-z+0.09=0

Solution for z^2-z+0.09=0 equation:

Simplifying
z2 + -1z + 0.09 = 0

Reorder the terms:
0.09 + -1z + z2 = 0

Solving
0.09 + -1z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-0.09' to each side of the equation.
0.09 + -1z + -0.09 + z2 = 0 + -0.09

Reorder the terms:
0.09 + -0.09 + -1z + z2 = 0 + -0.09

Combine like terms: 0.09 + -0.09 = 0.00
0.00 + -1z + z2 = 0 + -0.09
-1z + z2 = 0 + -0.09

Combine like terms: 0 + -0.09 = -0.09
-1z + z2 = -0.09

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = -0.09 + 0.25

Reorder the terms:
0.25 + -1z + z2 = -0.09 + 0.25

Combine like terms: -0.09 + 0.25 = 0.16
0.25 + -1z + z2 = 0.16

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 0.16

Calculate the square root of the right side: 0.4

Break this problem into two subproblems by setting 
(z + -0.5) equal to 0.4 and -0.4.

Subproblem 1
z + -0.5 = 0.4

Simplifying
z + -0.5 = 0.4

Reorder the terms:
-0.5 + z = 0.4

Solving
-0.5 + z = 0.4

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = 0.4 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = 0.4 + 0.5
z = 0.4 + 0.5

Combine like terms: 0.4 + 0.5 = 0.9
z = 0.9

Simplifying
z = 0.9

Subproblem 2
z + -0.5 = -0.4

Simplifying
z + -0.5 = -0.4

Reorder the terms:
-0.5 + z = -0.4

Solving
-0.5 + z = -0.4

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = -0.4 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = -0.4 + 0.5
z = -0.4 + 0.5

Combine like terms: -0.4 + 0.5 = 0.1
z = 0.1

Simplifying
z = 0.1

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.9, 0.1}