b=5b^2-6b+1

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Solution for b=5b^2-6b+1 equation:


Simplifying
b = 5b2 + -6b + 1

Reorder the terms:
b = 1 + -6b + 5b2

Solving
b = 1 + -6b + 5b2

Solving for variable 'b'.

Reorder the terms:
-1 + b + 6b + -5b2 = 1 + -6b + 5b2 + -1 + 6b + -5b2

Combine like terms: b + 6b = 7b
-1 + 7b + -5b2 = 1 + -6b + 5b2 + -1 + 6b + -5b2

Reorder the terms:
-1 + 7b + -5b2 = 1 + -1 + -6b + 6b + 5b2 + -5b2

Combine like terms: 1 + -1 = 0
-1 + 7b + -5b2 = 0 + -6b + 6b + 5b2 + -5b2
-1 + 7b + -5b2 = -6b + 6b + 5b2 + -5b2

Combine like terms: -6b + 6b = 0
-1 + 7b + -5b2 = 0 + 5b2 + -5b2
-1 + 7b + -5b2 = 5b2 + -5b2

Combine like terms: 5b2 + -5b2 = 0
-1 + 7b + -5b2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
0.2 + -1.4b + b2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -1.4b + -0.2 + b2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -1.4b + b2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -1.4b + b2 = 0 + -0.2
-1.4b + b2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-1.4b + b2 = -0.2

The b term is -1.4b.  Take half its coefficient (-0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
-1.4b + 0.49 + b2 = -0.2 + 0.49

Reorder the terms:
0.49 + -1.4b + b2 = -0.2 + 0.49

Combine like terms: -0.2 + 0.49 = 0.29
0.49 + -1.4b + b2 = 0.29

Factor a perfect square on the left side:
(b + -0.7)(b + -0.7) = 0.29

Calculate the square root of the right side: 0.538516481

Break this problem into two subproblems by setting 
(b + -0.7) equal to 0.538516481 and -0.538516481.

Subproblem 1

b + -0.7 = 0.538516481 Simplifying b + -0.7 = 0.538516481 Reorder the terms: -0.7 + b = 0.538516481 Solving -0.7 + b = 0.538516481 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + b = 0.538516481 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + b = 0.538516481 + 0.7 b = 0.538516481 + 0.7 Combine like terms: 0.538516481 + 0.7 = 1.238516481 b = 1.238516481 Simplifying b = 1.238516481

Subproblem 2

b + -0.7 = -0.538516481 Simplifying b + -0.7 = -0.538516481 Reorder the terms: -0.7 + b = -0.538516481 Solving -0.7 + b = -0.538516481 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + b = -0.538516481 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + b = -0.538516481 + 0.7 b = -0.538516481 + 0.7 Combine like terms: -0.538516481 + 0.7 = 0.161483519 b = 0.161483519 Simplifying b = 0.161483519

Solution

The solution to the problem is based on the solutions from the subproblems. b = {1.238516481, 0.161483519}

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