.5(b)(b-8)=10

Solution for .5(b)(b-8)=10 equation:

Simplifying
0.5(b)(b + -8) = 10

Reorder the terms:
0.5b(-8 + b) = 10
(-8 * 0.5b + b * 0.5b) = 10
(-4b + 0.5b2) = 10

Solving
-4b + 0.5b2 = 10

Solving for variable 'b'.

Reorder the terms:
-10 + -4b + 0.5b2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + -4b + 0.5b2 = 0

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

Divide each side by '0.5'.
-20 + -8b + b2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -8b + 20 + b2 = 0 + 20

Reorder the terms:
-20 + 20 + -8b + b2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -8b + b2 = 0 + 20
-8b + b2 = 0 + 20

Combine like terms: 0 + 20 = 20
-8b + b2 = 20

The b term is -8b.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8b + 16 + b2 = 20 + 16

Reorder the terms:
16 + -8b + b2 = 20 + 16

Combine like terms: 20 + 16 = 36
16 + -8b + b2 = 36

Factor a perfect square on the left side:
(b + -4)(b + -4) = 36

Calculate the square root of the right side: 6

Break this problem into two subproblems by setting 
(b + -4) equal to 6 and -6.

Subproblem 1
b + -4 = 6

Simplifying
b + -4 = 6

Reorder the terms:
-4 + b = 6

Solving
-4 + b = 6

Solving for variable 'b'.

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

Add '4' to each side of the equation.
-4 + 4 + b = 6 + 4

Combine like terms: -4 + 4 = 0
0 + b = 6 + 4
b = 6 + 4

Combine like terms: 6 + 4 = 10
b = 10

Simplifying
b = 10

Subproblem 2
b + -4 = -6

Simplifying
b + -4 = -6

Reorder the terms:
-4 + b = -6

Solving
-4 + b = -6

Solving for variable 'b'.

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

Add '4' to each side of the equation.
-4 + 4 + b = -6 + 4

Combine like terms: -4 + 4 = 0
0 + b = -6 + 4
b = -6 + 4

Combine like terms: -6 + 4 = -2
b = -2

Simplifying
b = -2

Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {10, -2}