.5(5b-10)(b)=420

Solution for .5(5b-10)(b)=420 equation:

Simplifying
0.5(5b + -10)(b) = 420

Reorder the terms:
0.5(-10 + 5b)(b) = 420

Reorder the terms for easier multiplication:
0.5b(-10 + 5b) = 420
(-10 * 0.5b + 5b * 0.5b) = 420
(-5b + 2.5b2) = 420

Solving
-5b + 2.5b2 = 420

Solving for variable 'b'.

Reorder the terms:
-420 + -5b + 2.5b2 = 420 + -420

Combine like terms: 420 + -420 = 0
-420 + -5b + 2.5b2 = 0

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

Divide each side by '2.5'.
-168 + -2b + b2 = 0

Move the constant term to the right:

Add '168' to each side of the equation.
-168 + -2b + 168 + b2 = 0 + 168

Reorder the terms:
-168 + 168 + -2b + b2 = 0 + 168

Combine like terms: -168 + 168 = 0
0 + -2b + b2 = 0 + 168
-2b + b2 = 0 + 168

Combine like terms: 0 + 168 = 168
-2b + b2 = 168

The b term is -2b.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2b + 1 + b2 = 168 + 1

Reorder the terms:
1 + -2b + b2 = 168 + 1

Combine like terms: 168 + 1 = 169
1 + -2b + b2 = 169

Factor a perfect square on the left side:
(b + -1)(b + -1) = 169

Calculate the square root of the right side: 13

Break this problem into two subproblems by setting 
(b + -1) equal to 13 and -13.

Subproblem 1
b + -1 = 13

Simplifying
b + -1 = 13

Reorder the terms:
-1 + b = 13

Solving
-1 + b = 13

Solving for variable 'b'.

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

Add '1' to each side of the equation.
-1 + 1 + b = 13 + 1

Combine like terms: -1 + 1 = 0
0 + b = 13 + 1
b = 13 + 1

Combine like terms: 13 + 1 = 14
b = 14

Simplifying
b = 14

Subproblem 2
b + -1 = -13

Simplifying
b + -1 = -13

Reorder the terms:
-1 + b = -13

Solving
-1 + b = -13

Solving for variable 'b'.

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

Add '1' to each side of the equation.
-1 + 1 + b = -13 + 1

Combine like terms: -1 + 1 = 0
0 + b = -13 + 1
b = -13 + 1

Combine like terms: -13 + 1 = -12
b = -12

Simplifying
b = -12

Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {14, -12}