3b^2-37b-90=0

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Solution for 3b^2-37b-90=0 equation:


Simplifying
3b2 + -37b + -90 = 0

Reorder the terms:
-90 + -37b + 3b2 = 0

Solving
-90 + -37b + 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'.
-30 + -12.33333333b + b2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + -12.33333333b + 30 + b2 = 0 + 30

Reorder the terms:
-30 + 30 + -12.33333333b + b2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + -12.33333333b + b2 = 0 + 30
-12.33333333b + b2 = 0 + 30

Combine like terms: 0 + 30 = 30
-12.33333333b + b2 = 30

The b term is -12.33333333b.  Take half its coefficient (-6.166666665).
Square it (38.02777776) and add it to both sides.

Add '38.02777776' to each side of the equation.
-12.33333333b + 38.02777776 + b2 = 30 + 38.02777776

Reorder the terms:
38.02777776 + -12.33333333b + b2 = 30 + 38.02777776

Combine like terms: 30 + 38.02777776 = 68.02777776
38.02777776 + -12.33333333b + b2 = 68.02777776

Factor a perfect square on the left side:
(b + -6.166666665)(b + -6.166666665) = 68.02777776

Calculate the square root of the right side: 8.247895353

Break this problem into two subproblems by setting 
(b + -6.166666665) equal to 8.247895353 and -8.247895353.

Subproblem 1

b + -6.166666665 = 8.247895353 Simplifying b + -6.166666665 = 8.247895353 Reorder the terms: -6.166666665 + b = 8.247895353 Solving -6.166666665 + b = 8.247895353 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + b = 8.247895353 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + b = 8.247895353 + 6.166666665 b = 8.247895353 + 6.166666665 Combine like terms: 8.247895353 + 6.166666665 = 14.414562018 b = 14.414562018 Simplifying b = 14.414562018

Subproblem 2

b + -6.166666665 = -8.247895353 Simplifying b + -6.166666665 = -8.247895353 Reorder the terms: -6.166666665 + b = -8.247895353 Solving -6.166666665 + b = -8.247895353 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + b = -8.247895353 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + b = -8.247895353 + 6.166666665 b = -8.247895353 + 6.166666665 Combine like terms: -8.247895353 + 6.166666665 = -2.081228688 b = -2.081228688 Simplifying b = -2.081228688

Solution

The solution to the problem is based on the solutions from the subproblems. b = {14.414562018, -2.081228688}

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