10b-45=3b(b-15)

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Solution for 10b-45=3b(b-15) equation:


Simplifying
10b + -45 = 3b(b + -15)

Reorder the terms:
-45 + 10b = 3b(b + -15)

Reorder the terms:
-45 + 10b = 3b(-15 + b)
-45 + 10b = (-15 * 3b + b * 3b)
-45 + 10b = (-45b + 3b2)

Solving
-45 + 10b = -45b + 3b2

Solving for variable 'b'.

Combine like terms: 10b + 45b = 55b
-45 + 55b + -3b2 = -45b + 3b2 + 45b + -3b2

Reorder the terms:
-45 + 55b + -3b2 = -45b + 45b + 3b2 + -3b2

Combine like terms: -45b + 45b = 0
-45 + 55b + -3b2 = 0 + 3b2 + -3b2
-45 + 55b + -3b2 = 3b2 + -3b2

Combine like terms: 3b2 + -3b2 = 0
-45 + 55b + -3b2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
15 + -18.33333333b + b2 = 0

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + -18.33333333b + -15 + b2 = 0 + -15

Reorder the terms:
15 + -15 + -18.33333333b + b2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + -18.33333333b + b2 = 0 + -15
-18.33333333b + b2 = 0 + -15

Combine like terms: 0 + -15 = -15
-18.33333333b + b2 = -15

The b term is -18.33333333b.  Take half its coefficient (-9.166666665).
Square it (84.02777775) and add it to both sides.

Add '84.02777775' to each side of the equation.
-18.33333333b + 84.02777775 + b2 = -15 + 84.02777775

Reorder the terms:
84.02777775 + -18.33333333b + b2 = -15 + 84.02777775

Combine like terms: -15 + 84.02777775 = 69.02777775
84.02777775 + -18.33333333b + b2 = 69.02777775

Factor a perfect square on the left side:
(b + -9.166666665)(b + -9.166666665) = 69.02777775

Calculate the square root of the right side: 8.308295719

Break this problem into two subproblems by setting 
(b + -9.166666665) equal to 8.308295719 and -8.308295719.

Subproblem 1

b + -9.166666665 = 8.308295719 Simplifying b + -9.166666665 = 8.308295719 Reorder the terms: -9.166666665 + b = 8.308295719 Solving -9.166666665 + b = 8.308295719 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '9.166666665' to each side of the equation. -9.166666665 + 9.166666665 + b = 8.308295719 + 9.166666665 Combine like terms: -9.166666665 + 9.166666665 = 0.000000000 0.000000000 + b = 8.308295719 + 9.166666665 b = 8.308295719 + 9.166666665 Combine like terms: 8.308295719 + 9.166666665 = 17.474962384 b = 17.474962384 Simplifying b = 17.474962384

Subproblem 2

b + -9.166666665 = -8.308295719 Simplifying b + -9.166666665 = -8.308295719 Reorder the terms: -9.166666665 + b = -8.308295719 Solving -9.166666665 + b = -8.308295719 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '9.166666665' to each side of the equation. -9.166666665 + 9.166666665 + b = -8.308295719 + 9.166666665 Combine like terms: -9.166666665 + 9.166666665 = 0.000000000 0.000000000 + b = -8.308295719 + 9.166666665 b = -8.308295719 + 9.166666665 Combine like terms: -8.308295719 + 9.166666665 = 0.858370946 b = 0.858370946 Simplifying b = 0.858370946

Solution

The solution to the problem is based on the solutions from the subproblems. b = {17.474962384, 0.858370946}

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