10z(1-z)=.09

Solution for 10z(1-z)=.09 equation:

Simplifying
10z(1 + -1z) = 0.09
(1 * 10z + -1z * 10z) = 0.09
(10z + -10z2) = 0.09

Solving
10z + -10z2 = 0.09

Solving for variable 'z'.

Reorder the terms:
-0.09 + 10z + -10z2 = 0.09 + -0.09

Combine like terms: 0.09 + -0.09 = 0.00
-0.09 + 10z + -10z2 = 0.00

Begin completing the square.  Divide all terms by
-10 the coefficient of the squared term: 

Divide each side by '-10'.
0.009 + -1z + z2 = 0.000

Move the constant term to the right:

Add '-0.009' to each side of the equation.
0.009 + -1z + -0.009 + z2 = 0.000 + -0.009

Reorder the terms:
0.009 + -0.009 + -1z + z2 = 0.000 + -0.009

Combine like terms: 0.009 + -0.009 = 0.000
0.000 + -1z + z2 = 0.000 + -0.009
-1z + z2 = 0.000 + -0.009

Combine like terms: 0.000 + -0.009 = -0.009
-1z + z2 = -0.009

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 = -0.009 + 0.25

Reorder the terms:
0.25 + -1z + z2 = -0.009 + 0.25

Combine like terms: -0.009 + 0.25 = 0.241
0.25 + -1z + z2 = 0.241

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 0.241

Calculate the square root of the right side: 0.490917508

Break this problem into two subproblems by setting 
(z + -0.5) equal to 0.490917508 and -0.490917508.

Subproblem 1
z + -0.5 = 0.490917508

Simplifying
z + -0.5 = 0.490917508

Reorder the terms:
-0.5 + z = 0.490917508

Solving
-0.5 + z = 0.490917508

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 = 0.490917508 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = 0.490917508 + 0.5
z = 0.490917508 + 0.5

Combine like terms: 0.490917508 + 0.5 = 0.990917508
z = 0.990917508

Simplifying
z = 0.990917508

Subproblem 2
z + -0.5 = -0.490917508

Simplifying
z + -0.5 = -0.490917508

Reorder the terms:
-0.5 + z = -0.490917508

Solving
-0.5 + z = -0.490917508

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 = -0.490917508 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = -0.490917508 + 0.5
z = -0.490917508 + 0.5

Combine like terms: -0.490917508 + 0.5 = 0.009082492
z = 0.009082492

Simplifying
z = 0.009082492

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.990917508, 0.009082492}