0=13b^2-60b+21

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Solution for 0=13b^2-60b+21 equation:


Simplifying
0 = 13b2 + -60b + 21

Reorder the terms:
0 = 21 + -60b + 13b2

Solving
0 = 21 + -60b + 13b2

Solving for variable 'b'.

Combine like terms: 0 + -21 = -21
-21 + 60b + -13b2 = 21 + -60b + 13b2 + -21 + 60b + -13b2

Reorder the terms:
-21 + 60b + -13b2 = 21 + -21 + -60b + 60b + 13b2 + -13b2

Combine like terms: 21 + -21 = 0
-21 + 60b + -13b2 = 0 + -60b + 60b + 13b2 + -13b2
-21 + 60b + -13b2 = -60b + 60b + 13b2 + -13b2

Combine like terms: -60b + 60b = 0
-21 + 60b + -13b2 = 0 + 13b2 + -13b2
-21 + 60b + -13b2 = 13b2 + -13b2

Combine like terms: 13b2 + -13b2 = 0
-21 + 60b + -13b2 = 0

Begin completing the square.  Divide all terms by
-13 the coefficient of the squared term: 

Divide each side by '-13'.
1.615384615 + -4.615384615b + b2 = 0

Move the constant term to the right:

Add '-1.615384615' to each side of the equation.
1.615384615 + -4.615384615b + -1.615384615 + b2 = 0 + -1.615384615

Reorder the terms:
1.615384615 + -1.615384615 + -4.615384615b + b2 = 0 + -1.615384615

Combine like terms: 1.615384615 + -1.615384615 = 0.000000000
0.000000000 + -4.615384615b + b2 = 0 + -1.615384615
-4.615384615b + b2 = 0 + -1.615384615

Combine like terms: 0 + -1.615384615 = -1.615384615
-4.615384615b + b2 = -1.615384615

The b term is -4.615384615b.  Take half its coefficient (-2.307692308).
Square it (5.325443788) and add it to both sides.

Add '5.325443788' to each side of the equation.
-4.615384615b + 5.325443788 + b2 = -1.615384615 + 5.325443788

Reorder the terms:
5.325443788 + -4.615384615b + b2 = -1.615384615 + 5.325443788

Combine like terms: -1.615384615 + 5.325443788 = 3.710059173
5.325443788 + -4.615384615b + b2 = 3.710059173

Factor a perfect square on the left side:
(b + -2.307692308)(b + -2.307692308) = 3.710059173

Calculate the square root of the right side: 1.926151389

Break this problem into two subproblems by setting 
(b + -2.307692308) equal to 1.926151389 and -1.926151389.

Subproblem 1

b + -2.307692308 = 1.926151389 Simplifying b + -2.307692308 = 1.926151389 Reorder the terms: -2.307692308 + b = 1.926151389 Solving -2.307692308 + b = 1.926151389 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + b = 1.926151389 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + b = 1.926151389 + 2.307692308 b = 1.926151389 + 2.307692308 Combine like terms: 1.926151389 + 2.307692308 = 4.233843697 b = 4.233843697 Simplifying b = 4.233843697

Subproblem 2

b + -2.307692308 = -1.926151389 Simplifying b + -2.307692308 = -1.926151389 Reorder the terms: -2.307692308 + b = -1.926151389 Solving -2.307692308 + b = -1.926151389 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + b = -1.926151389 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + b = -1.926151389 + 2.307692308 b = -1.926151389 + 2.307692308 Combine like terms: -1.926151389 + 2.307692308 = 0.381540919 b = 0.381540919 Simplifying b = 0.381540919

Solution

The solution to the problem is based on the solutions from the subproblems. b = {4.233843697, 0.381540919}

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