6b^2+32b+15=0

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Solution for 6b^2+32b+15=0 equation: 


Simplifying
6b2 + 32b + 15 = 0

Reorder the terms:
15 + 32b + 6b2 = 0

Solving
15 + 32b + 6b2 = 0

Solving for variable 'b'.

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

Divide each side by '6'.
2.5 + 5.333333333b + b2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 5.333333333b + -2.5 + b2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 5.333333333b + b2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 5.333333333b + b2 = 0 + -2.5
5.333333333b + b2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
5.333333333b + b2 = -2.5

The b term is 5.333333333b.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333b + 7.111111113 + b2 = -2.5 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333b + b2 = -2.5 + 7.111111113

Combine like terms: -2.5 + 7.111111113 = 4.611111113
7.111111113 + 5.333333333b + b2 = 4.611111113

Factor a perfect square on the left side:
(b + 2.666666667)(b + 2.666666667) = 4.611111113

Calculate the square root of the right side: 2.147349788

Break this problem into two subproblems by setting 
(b + 2.666666667) equal to 2.147349788 and -2.147349788.


Subproblem 1
b + 2.666666667 = 2.147349788

Simplifying
b + 2.666666667 = 2.147349788

Reorder the terms:
2.666666667 + b = 2.147349788

Solving
2.666666667 + b = 2.147349788

Solving for variable 'b'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + b = 2.147349788 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + b = 2.147349788 + -2.666666667
b = 2.147349788 + -2.666666667

Combine like terms: 2.147349788 + -2.666666667 = -0.519316879
b = -0.519316879

Simplifying
b = -0.519316879


Subproblem 2
b + 2.666666667 = -2.147349788

Simplifying
b + 2.666666667 = -2.147349788

Reorder the terms:
2.666666667 + b = -2.147349788

Solving
2.666666667 + b = -2.147349788

Solving for variable 'b'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + b = -2.147349788 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + b = -2.147349788 + -2.666666667
b = -2.147349788 + -2.666666667

Combine like terms: -2.147349788 + -2.666666667 = -4.814016455
b = -4.814016455

Simplifying
b = -4.814016455


Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {-0.519316879, -4.814016455}

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