10b^2=27b-1

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Solution for 10b^2=27b-1 equation:


Simplifying
10b2 = 27b + -1

Reorder the terms:
10b2 = -1 + 27b

Solving
10b2 = -1 + 27b

Solving for variable 'b'.

Reorder the terms:
1 + -27b + 10b2 = -1 + 27b + 1 + -27b

Reorder the terms:
1 + -27b + 10b2 = -1 + 1 + 27b + -27b

Combine like terms: -1 + 1 = 0
1 + -27b + 10b2 = 0 + 27b + -27b
1 + -27b + 10b2 = 27b + -27b

Combine like terms: 27b + -27b = 0
1 + -27b + 10b2 = 0

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

Divide each side by '10'.
0.1 + -2.7b + b2 = 0

Move the constant term to the right:

Add '-0.1' to each side of the equation.
0.1 + -2.7b + -0.1 + b2 = 0 + -0.1

Reorder the terms:
0.1 + -0.1 + -2.7b + b2 = 0 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + -2.7b + b2 = 0 + -0.1
-2.7b + b2 = 0 + -0.1

Combine like terms: 0 + -0.1 = -0.1
-2.7b + b2 = -0.1

The b term is -2.7b.  Take half its coefficient (-1.35).
Square it (1.8225) and add it to both sides.

Add '1.8225' to each side of the equation.
-2.7b + 1.8225 + b2 = -0.1 + 1.8225

Reorder the terms:
1.8225 + -2.7b + b2 = -0.1 + 1.8225

Combine like terms: -0.1 + 1.8225 = 1.7225
1.8225 + -2.7b + b2 = 1.7225

Factor a perfect square on the left side:
(b + -1.35)(b + -1.35) = 1.7225

Calculate the square root of the right side: 1.312440475

Break this problem into two subproblems by setting 
(b + -1.35) equal to 1.312440475 and -1.312440475.

Subproblem 1

b + -1.35 = 1.312440475 Simplifying b + -1.35 = 1.312440475 Reorder the terms: -1.35 + b = 1.312440475 Solving -1.35 + b = 1.312440475 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + b = 1.312440475 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + b = 1.312440475 + 1.35 b = 1.312440475 + 1.35 Combine like terms: 1.312440475 + 1.35 = 2.662440475 b = 2.662440475 Simplifying b = 2.662440475

Subproblem 2

b + -1.35 = -1.312440475 Simplifying b + -1.35 = -1.312440475 Reorder the terms: -1.35 + b = -1.312440475 Solving -1.35 + b = -1.312440475 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.35' to each side of the equation. -1.35 + 1.35 + b = -1.312440475 + 1.35 Combine like terms: -1.35 + 1.35 = 0.00 0.00 + b = -1.312440475 + 1.35 b = -1.312440475 + 1.35 Combine like terms: -1.312440475 + 1.35 = 0.037559525 b = 0.037559525 Simplifying b = 0.037559525

Solution

The solution to the problem is based on the solutions from the subproblems. b = {2.662440475, 0.037559525}

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