7b^2-20b-31=0

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Solution for 7b^2-20b-31=0 equation: 


Simplifying
7b2 + -20b + -31 = 0

Reorder the terms:
-31 + -20b + 7b2 = 0

Solving
-31 + -20b + 7b2 = 0

Solving for variable 'b'.

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

Divide each side by '7'.
-4.428571429 + -2.857142857b + b2 = 0

Move the constant term to the right:

Add '4.428571429' to each side of the equation.
-4.428571429 + -2.857142857b + 4.428571429 + b2 = 0 + 4.428571429

Reorder the terms:
-4.428571429 + 4.428571429 + -2.857142857b + b2 = 0 + 4.428571429

Combine like terms: -4.428571429 + 4.428571429 = 0.000000000
0.000000000 + -2.857142857b + b2 = 0 + 4.428571429
-2.857142857b + b2 = 0 + 4.428571429

Combine like terms: 0 + 4.428571429 = 4.428571429
-2.857142857b + b2 = 4.428571429

The b term is -2.857142857b.  Take half its coefficient (-1.428571429).
Square it (2.040816328) and add it to both sides.

Add '2.040816328' to each side of the equation.
-2.857142857b + 2.040816328 + b2 = 4.428571429 + 2.040816328

Reorder the terms:
2.040816328 + -2.857142857b + b2 = 4.428571429 + 2.040816328

Combine like terms: 4.428571429 + 2.040816328 = 6.469387757
2.040816328 + -2.857142857b + b2 = 6.469387757

Factor a perfect square on the left side:
(b + -1.428571429)(b + -1.428571429) = 6.469387757

Calculate the square root of the right side: 2.543499117

Break this problem into two subproblems by setting 
(b + -1.428571429) equal to 2.543499117 and -2.543499117.


Subproblem 1
b + -1.428571429 = 2.543499117

Simplifying
b + -1.428571429 = 2.543499117

Reorder the terms:
-1.428571429 + b = 2.543499117

Solving
-1.428571429 + b = 2.543499117

Solving for variable 'b'.

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

Add '1.428571429' to each side of the equation.
-1.428571429 + 1.428571429 + b = 2.543499117 + 1.428571429

Combine like terms: -1.428571429 + 1.428571429 = 0.000000000
0.000000000 + b = 2.543499117 + 1.428571429
b = 2.543499117 + 1.428571429

Combine like terms: 2.543499117 + 1.428571429 = 3.972070546
b = 3.972070546

Simplifying
b = 3.972070546


Subproblem 2
b + -1.428571429 = -2.543499117

Simplifying
b + -1.428571429 = -2.543499117

Reorder the terms:
-1.428571429 + b = -2.543499117

Solving
-1.428571429 + b = -2.543499117

Solving for variable 'b'.

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

Add '1.428571429' to each side of the equation.
-1.428571429 + 1.428571429 + b = -2.543499117 + 1.428571429

Combine like terms: -1.428571429 + 1.428571429 = 0.000000000
0.000000000 + b = -2.543499117 + 1.428571429
b = -2.543499117 + 1.428571429

Combine like terms: -2.543499117 + 1.428571429 = -1.114927688
b = -1.114927688

Simplifying
b = -1.114927688


Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {3.972070546, -1.114927688}

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