6b^2=6b+23

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Solution for 6b^2=6b+23 equation:


Simplifying
6b2 = 6b + 23

Reorder the terms:
6b2 = 23 + 6b

Solving
6b2 = 23 + 6b

Solving for variable 'b'.

Reorder the terms:
-23 + -6b + 6b2 = 23 + 6b + -23 + -6b

Reorder the terms:
-23 + -6b + 6b2 = 23 + -23 + 6b + -6b

Combine like terms: 23 + -23 = 0
-23 + -6b + 6b2 = 0 + 6b + -6b
-23 + -6b + 6b2 = 6b + -6b

Combine like terms: 6b + -6b = 0
-23 + -6b + 6b2 = 0

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

Divide each side by '6'.
-3.833333333 + -1b + b2 = 0

Move the constant term to the right:

Add '3.833333333' to each side of the equation.
-3.833333333 + -1b + 3.833333333 + b2 = 0 + 3.833333333

Reorder the terms:
-3.833333333 + 3.833333333 + -1b + b2 = 0 + 3.833333333

Combine like terms: -3.833333333 + 3.833333333 = 0.000000000
0.000000000 + -1b + b2 = 0 + 3.833333333
-1b + b2 = 0 + 3.833333333

Combine like terms: 0 + 3.833333333 = 3.833333333
-1b + b2 = 3.833333333

The b term is -1b.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1b + 0.25 + b2 = 3.833333333 + 0.25

Reorder the terms:
0.25 + -1b + b2 = 3.833333333 + 0.25

Combine like terms: 3.833333333 + 0.25 = 4.083333333
0.25 + -1b + b2 = 4.083333333

Factor a perfect square on the left side:
(b + -0.5)(b + -0.5) = 4.083333333

Calculate the square root of the right side: 2.020725942

Break this problem into two subproblems by setting 
(b + -0.5) equal to 2.020725942 and -2.020725942.

Subproblem 1

b + -0.5 = 2.020725942 Simplifying b + -0.5 = 2.020725942 Reorder the terms: -0.5 + b = 2.020725942 Solving -0.5 + b = 2.020725942 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + b = 2.020725942 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 2.020725942 + 0.5 b = 2.020725942 + 0.5 Combine like terms: 2.020725942 + 0.5 = 2.520725942 b = 2.520725942 Simplifying b = 2.520725942

Subproblem 2

b + -0.5 = -2.020725942 Simplifying b + -0.5 = -2.020725942 Reorder the terms: -0.5 + b = -2.020725942 Solving -0.5 + b = -2.020725942 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + b = -2.020725942 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -2.020725942 + 0.5 b = -2.020725942 + 0.5 Combine like terms: -2.020725942 + 0.5 = -1.520725942 b = -1.520725942 Simplifying b = -1.520725942

Solution

The solution to the problem is based on the solutions from the subproblems. b = {2.520725942, -1.520725942}

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