(1-b)*b=0.09

Solution for (1-b)*b=0.09 equation:

Simplifying
(1 + -1b) * b = 0.09

Reorder the terms for easier multiplication:
b(1 + -1b) = 0.09
(1 * b + -1b * b) = 0.09
(1b + -1b2) = 0.09

Solving
1b + -1b2 = 0.09

Solving for variable 'b'.

Reorder the terms:
-0.09 + 1b + -1b2 = 0.09 + -0.09

Combine like terms: 0.09 + -0.09 = 0.00
-0.09 + 1b + -1b2 = 0.00

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
0.09 + -1b + b2 = 0.00

Move the constant term to the right:

Add '-0.09' to each side of the equation.
0.09 + -1b + -0.09 + b2 = 0.00 + -0.09

Reorder the terms:
0.09 + -0.09 + -1b + b2 = 0.00 + -0.09

Combine like terms: 0.09 + -0.09 = 0.00
0.00 + -1b + b2 = 0.00 + -0.09
-1b + b2 = 0.00 + -0.09

Combine like terms: 0.00 + -0.09 = -0.09
-1b + b2 = -0.09

The b term is -1b.  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.
-1b + 0.25 + b2 = -0.09 + 0.25

Reorder the terms:
0.25 + -1b + b2 = -0.09 + 0.25

Combine like terms: -0.09 + 0.25 = 0.16
0.25 + -1b + b2 = 0.16

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

Calculate the square root of the right side: 0.4

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

Subproblem 1
b + -0.5 = 0.4

Simplifying
b + -0.5 = 0.4

Reorder the terms:
-0.5 + b = 0.4

Solving
-0.5 + b = 0.4

Solving for variable 'b'.

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

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

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

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

Simplifying
b = 0.9

Subproblem 2
b + -0.5 = -0.4

Simplifying
b + -0.5 = -0.4

Reorder the terms:
-0.5 + b = -0.4

Solving
-0.5 + b = -0.4

Solving for variable 'b'.

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

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

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

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

Simplifying
b = 0.1

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