38=5b(2b-3)+3b+1

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Solution for 38=5b(2b-3)+3b+1 equation:


Simplifying
38 = 5b(2b + -3) + 3b + 1

Reorder the terms:
38 = 5b(-3 + 2b) + 3b + 1
38 = (-3 * 5b + 2b * 5b) + 3b + 1
38 = (-15b + 10b2) + 3b + 1

Reorder the terms:
38 = 1 + -15b + 3b + 10b2

Combine like terms: -15b + 3b = -12b
38 = 1 + -12b + 10b2

Solving
38 = 1 + -12b + 10b2

Solving for variable 'b'.

Combine like terms: 38 + -1 = 37
37 + 12b + -10b2 = 1 + -12b + 10b2 + -1 + 12b + -10b2

Reorder the terms:
37 + 12b + -10b2 = 1 + -1 + -12b + 12b + 10b2 + -10b2

Combine like terms: 1 + -1 = 0
37 + 12b + -10b2 = 0 + -12b + 12b + 10b2 + -10b2
37 + 12b + -10b2 = -12b + 12b + 10b2 + -10b2

Combine like terms: -12b + 12b = 0
37 + 12b + -10b2 = 0 + 10b2 + -10b2
37 + 12b + -10b2 = 10b2 + -10b2

Combine like terms: 10b2 + -10b2 = 0
37 + 12b + -10b2 = 0

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

Divide each side by '-10'.
-3.7 + -1.2b + b2 = 0

Move the constant term to the right:

Add '3.7' to each side of the equation.
-3.7 + -1.2b + 3.7 + b2 = 0 + 3.7

Reorder the terms:
-3.7 + 3.7 + -1.2b + b2 = 0 + 3.7

Combine like terms: -3.7 + 3.7 = 0.0
0.0 + -1.2b + b2 = 0 + 3.7
-1.2b + b2 = 0 + 3.7

Combine like terms: 0 + 3.7 = 3.7
-1.2b + b2 = 3.7

The b term is -1.2b.  Take half its coefficient (-0.6).
Square it (0.36) and add it to both sides.

Add '0.36' to each side of the equation.
-1.2b + 0.36 + b2 = 3.7 + 0.36

Reorder the terms:
0.36 + -1.2b + b2 = 3.7 + 0.36

Combine like terms: 3.7 + 0.36 = 4.06
0.36 + -1.2b + b2 = 4.06

Factor a perfect square on the left side:
(b + -0.6)(b + -0.6) = 4.06

Calculate the square root of the right side: 2.014944168

Break this problem into two subproblems by setting 
(b + -0.6) equal to 2.014944168 and -2.014944168.

Subproblem 1

b + -0.6 = 2.014944168 Simplifying b + -0.6 = 2.014944168 Reorder the terms: -0.6 + b = 2.014944168 Solving -0.6 + b = 2.014944168 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.6' to each side of the equation. -0.6 + 0.6 + b = 2.014944168 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + b = 2.014944168 + 0.6 b = 2.014944168 + 0.6 Combine like terms: 2.014944168 + 0.6 = 2.614944168 b = 2.614944168 Simplifying b = 2.614944168

Subproblem 2

b + -0.6 = -2.014944168 Simplifying b + -0.6 = -2.014944168 Reorder the terms: -0.6 + b = -2.014944168 Solving -0.6 + b = -2.014944168 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.6' to each side of the equation. -0.6 + 0.6 + b = -2.014944168 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + b = -2.014944168 + 0.6 b = -2.014944168 + 0.6 Combine like terms: -2.014944168 + 0.6 = -1.414944168 b = -1.414944168 Simplifying b = -1.414944168

Solution

The solution to the problem is based on the solutions from the subproblems. b = {2.614944168, -1.414944168}

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