3b(b+4)=24-2(b-3)

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Solution for 3b(b+4)=24-2(b-3) equation:


Simplifying
3b(b + 4) = 24 + -2(b + -3)

Reorder the terms:
3b(4 + b) = 24 + -2(b + -3)
(4 * 3b + b * 3b) = 24 + -2(b + -3)
(12b + 3b2) = 24 + -2(b + -3)

Reorder the terms:
12b + 3b2 = 24 + -2(-3 + b)
12b + 3b2 = 24 + (-3 * -2 + b * -2)
12b + 3b2 = 24 + (6 + -2b)

Combine like terms: 24 + 6 = 30
12b + 3b2 = 30 + -2b

Solving
12b + 3b2 = 30 + -2b

Solving for variable 'b'.

Reorder the terms:
-30 + 12b + 2b + 3b2 = 30 + -2b + -30 + 2b

Combine like terms: 12b + 2b = 14b
-30 + 14b + 3b2 = 30 + -2b + -30 + 2b

Reorder the terms:
-30 + 14b + 3b2 = 30 + -30 + -2b + 2b

Combine like terms: 30 + -30 = 0
-30 + 14b + 3b2 = 0 + -2b + 2b
-30 + 14b + 3b2 = -2b + 2b

Combine like terms: -2b + 2b = 0
-30 + 14b + 3b2 = 0

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

Divide each side by '3'.
-10 + 4.666666667b + b2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 4.666666667b + 10 + b2 = 0 + 10

Reorder the terms:
-10 + 10 + 4.666666667b + b2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 4.666666667b + b2 = 0 + 10
4.666666667b + b2 = 0 + 10

Combine like terms: 0 + 10 = 10
4.666666667b + b2 = 10

The b term is 4.666666667b.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667b + 5.444444448 + b2 = 10 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667b + b2 = 10 + 5.444444448

Combine like terms: 10 + 5.444444448 = 15.444444448
5.444444448 + 4.666666667b + b2 = 15.444444448

Factor a perfect square on the left side:
(b + 2.333333334)(b + 2.333333334) = 15.444444448

Calculate the square root of the right side: 3.929942041

Break this problem into two subproblems by setting 
(b + 2.333333334) equal to 3.929942041 and -3.929942041.

Subproblem 1

b + 2.333333334 = 3.929942041 Simplifying b + 2.333333334 = 3.929942041 Reorder the terms: 2.333333334 + b = 3.929942041 Solving 2.333333334 + b = 3.929942041 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + b = 3.929942041 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + b = 3.929942041 + -2.333333334 b = 3.929942041 + -2.333333334 Combine like terms: 3.929942041 + -2.333333334 = 1.596608707 b = 1.596608707 Simplifying b = 1.596608707

Subproblem 2

b + 2.333333334 = -3.929942041 Simplifying b + 2.333333334 = -3.929942041 Reorder the terms: 2.333333334 + b = -3.929942041 Solving 2.333333334 + b = -3.929942041 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + b = -3.929942041 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + b = -3.929942041 + -2.333333334 b = -3.929942041 + -2.333333334 Combine like terms: -3.929942041 + -2.333333334 = -6.263275375 b = -6.263275375 Simplifying b = -6.263275375

Solution

The solution to the problem is based on the solutions from the subproblems. b = {1.596608707, -6.263275375}

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