.5(b^2-5b)=42

Solution for .5(b^2-5b)=42 equation:

Simplifying
0.5(b2 + -5b) = 42

Reorder the terms:
0.5(-5b + b2) = 42
(-5b * 0.5 + b2 * 0.5) = 42
(-2.5b + 0.5b2) = 42

Solving
-2.5b + 0.5b2 = 42

Solving for variable 'b'.

Reorder the terms:
-42 + -2.5b + 0.5b2 = 42 + -42

Combine like terms: 42 + -42 = 0
-42 + -2.5b + 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'.
-84 + -5b + b2 = 0

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + -5b + 84 + b2 = 0 + 84

Reorder the terms:
-84 + 84 + -5b + b2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + -5b + b2 = 0 + 84
-5b + b2 = 0 + 84

Combine like terms: 0 + 84 = 84
-5b + b2 = 84

The b term is -5b.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5b + 6.25 + b2 = 84 + 6.25

Reorder the terms:
6.25 + -5b + b2 = 84 + 6.25

Combine like terms: 84 + 6.25 = 90.25
6.25 + -5b + b2 = 90.25

Factor a perfect square on the left side:
(b + -2.5)(b + -2.5) = 90.25

Calculate the square root of the right side: 9.5

Break this problem into two subproblems by setting 
(b + -2.5) equal to 9.5 and -9.5.

Subproblem 1
b + -2.5 = 9.5

Simplifying
b + -2.5 = 9.5

Reorder the terms:
-2.5 + b = 9.5

Solving
-2.5 + b = 9.5

Solving for variable 'b'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + b = 9.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + b = 9.5 + 2.5
b = 9.5 + 2.5

Combine like terms: 9.5 + 2.5 = 12
b = 12

Simplifying
b = 12

Subproblem 2
b + -2.5 = -9.5

Simplifying
b + -2.5 = -9.5

Reorder the terms:
-2.5 + b = -9.5

Solving
-2.5 + b = -9.5

Solving for variable 'b'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + b = -9.5 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + b = -9.5 + 2.5
b = -9.5 + 2.5

Combine like terms: -9.5 + 2.5 = -7
b = -7

Simplifying
b = -7

Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {12, -7}