b(6b+7)=8

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Solution for b(6b+7)=8 equation:


Simplifying
b(6b + 7) = 8

Reorder the terms:
b(7 + 6b) = 8
(7 * b + 6b * b) = 8
(7b + 6b2) = 8

Solving
7b + 6b2 = 8

Solving for variable 'b'.

Reorder the terms:
-8 + 7b + 6b2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 7b + 6b2 = 0

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

Divide each side by '6'.
-1.333333333 + 1.166666667b + b2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 1.166666667b + 1.333333333 + b2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 1.166666667b + b2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 1.166666667b + b2 = 0 + 1.333333333
1.166666667b + b2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
1.166666667b + b2 = 1.333333333

The b term is 1.166666667b.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667b + 0.3402777780 + b2 = 1.333333333 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667b + b2 = 1.333333333 + 0.3402777780

Combine like terms: 1.333333333 + 0.3402777780 = 1.673611111
0.3402777780 + 1.166666667b + b2 = 1.673611111

Factor a perfect square on the left side:
(b + 0.5833333335)(b + 0.5833333335) = 1.673611111

Calculate the square root of the right side: 1.293681225

Break this problem into two subproblems by setting 
(b + 0.5833333335) equal to 1.293681225 and -1.293681225.

Subproblem 1

b + 0.5833333335 = 1.293681225 Simplifying b + 0.5833333335 = 1.293681225 Reorder the terms: 0.5833333335 + b = 1.293681225 Solving 0.5833333335 + b = 1.293681225 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + b = 1.293681225 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + b = 1.293681225 + -0.5833333335 b = 1.293681225 + -0.5833333335 Combine like terms: 1.293681225 + -0.5833333335 = 0.7103478915 b = 0.7103478915 Simplifying b = 0.7103478915

Subproblem 2

b + 0.5833333335 = -1.293681225 Simplifying b + 0.5833333335 = -1.293681225 Reorder the terms: 0.5833333335 + b = -1.293681225 Solving 0.5833333335 + b = -1.293681225 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + b = -1.293681225 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + b = -1.293681225 + -0.5833333335 b = -1.293681225 + -0.5833333335 Combine like terms: -1.293681225 + -0.5833333335 = -1.8770145585 b = -1.8770145585 Simplifying b = -1.8770145585

Solution

The solution to the problem is based on the solutions from the subproblems. b = {0.7103478915, -1.8770145585}

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