3b^2+16b-27=0

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Solution for 3b^2+16b-27=0 equation: 


Simplifying
3b2 + 16b + -27 = 0

Reorder the terms:
-27 + 16b + 3b2 = 0

Solving
-27 + 16b + 3b2 = 0

Solving for variable 'b'.

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

Divide each side by '3'.
-9 + 5.333333333b + b2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 5.333333333b + 9 + b2 = 0 + 9

Reorder the terms:
-9 + 9 + 5.333333333b + b2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 5.333333333b + b2 = 0 + 9
5.333333333b + b2 = 0 + 9

Combine like terms: 0 + 9 = 9
5.333333333b + b2 = 9

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 = 9 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333b + b2 = 9 + 7.111111113

Combine like terms: 9 + 7.111111113 = 16.111111113
7.111111113 + 5.333333333b + b2 = 16.111111113

Factor a perfect square on the left side:
(b + 2.666666667)(b + 2.666666667) = 16.111111113

Calculate the square root of the right side: 4.01386486

Break this problem into two subproblems by setting 
(b + 2.666666667) equal to 4.01386486 and -4.01386486.


Subproblem 1
b + 2.666666667 = 4.01386486

Simplifying
b + 2.666666667 = 4.01386486

Reorder the terms:
2.666666667 + b = 4.01386486

Solving
2.666666667 + b = 4.01386486

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 = 4.01386486 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + b = 4.01386486 + -2.666666667
b = 4.01386486 + -2.666666667

Combine like terms: 4.01386486 + -2.666666667 = 1.347198193
b = 1.347198193

Simplifying
b = 1.347198193


Subproblem 2
b + 2.666666667 = -4.01386486

Simplifying
b + 2.666666667 = -4.01386486

Reorder the terms:
2.666666667 + b = -4.01386486

Solving
2.666666667 + b = -4.01386486

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 = -4.01386486 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + b = -4.01386486 + -2.666666667
b = -4.01386486 + -2.666666667

Combine like terms: -4.01386486 + -2.666666667 = -6.680531527
b = -6.680531527

Simplifying
b = -6.680531527


Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {1.347198193, -6.680531527}

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