1.3b(8-b)+3.7b=-5.2

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Solution for 1.3b(8-b)+3.7b=-5.2 equation:


Simplifying
1.3b(8 + -1b) + 3.7b = -5.2
(8 * 1.3b + -1b * 1.3b) + 3.7b = -5.2
(10.4b + -1.3b2) + 3.7b = -5.2

Reorder the terms:
10.4b + 3.7b + -1.3b2 = -5.2

Combine like terms: 10.4b + 3.7b = 14.1b
14.1b + -1.3b2 = -5.2

Solving
14.1b + -1.3b2 = -5.2

Solving for variable 'b'.

Reorder the terms:
5.2 + 14.1b + -1.3b2 = -5.2 + 5.2

Combine like terms: -5.2 + 5.2 = 0.0
5.2 + 14.1b + -1.3b2 = 0.0

Begin completing the square.  Divide all terms by
-1.3 the coefficient of the squared term: 

Divide each side by '-1.3'.
-4 + -10.84615385b + b2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -10.84615385b + 4 + b2 = 0 + 4

Reorder the terms:
-4 + 4 + -10.84615385b + b2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -10.84615385b + b2 = 0 + 4
-10.84615385b + b2 = 0 + 4

Combine like terms: 0 + 4 = 4
-10.84615385b + b2 = 4

The b term is -10.84615385b.  Take half its coefficient (-5.423076925).
Square it (29.40976333) and add it to both sides.

Add '29.40976333' to each side of the equation.
-10.84615385b + 29.40976333 + b2 = 4 + 29.40976333

Reorder the terms:
29.40976333 + -10.84615385b + b2 = 4 + 29.40976333

Combine like terms: 4 + 29.40976333 = 33.40976333
29.40976333 + -10.84615385b + b2 = 33.40976333

Factor a perfect square on the left side:
(b + -5.423076925)(b + -5.423076925) = 33.40976333

Calculate the square root of the right side: 5.780117934

Break this problem into two subproblems by setting 
(b + -5.423076925) equal to 5.780117934 and -5.780117934.

Subproblem 1

b + -5.423076925 = 5.780117934 Simplifying b + -5.423076925 = 5.780117934 Reorder the terms: -5.423076925 + b = 5.780117934 Solving -5.423076925 + b = 5.780117934 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '5.423076925' to each side of the equation. -5.423076925 + 5.423076925 + b = 5.780117934 + 5.423076925 Combine like terms: -5.423076925 + 5.423076925 = 0.000000000 0.000000000 + b = 5.780117934 + 5.423076925 b = 5.780117934 + 5.423076925 Combine like terms: 5.780117934 + 5.423076925 = 11.203194859 b = 11.203194859 Simplifying b = 11.203194859

Subproblem 2

b + -5.423076925 = -5.780117934 Simplifying b + -5.423076925 = -5.780117934 Reorder the terms: -5.423076925 + b = -5.780117934 Solving -5.423076925 + b = -5.780117934 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '5.423076925' to each side of the equation. -5.423076925 + 5.423076925 + b = -5.780117934 + 5.423076925 Combine like terms: -5.423076925 + 5.423076925 = 0.000000000 0.000000000 + b = -5.780117934 + 5.423076925 b = -5.780117934 + 5.423076925 Combine like terms: -5.780117934 + 5.423076925 = -0.357041009 b = -0.357041009 Simplifying b = -0.357041009

Solution

The solution to the problem is based on the solutions from the subproblems. b = {11.203194859, -0.357041009}

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