(9b)(9-b)=32

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Solution for (9b)(9-b)=32 equation:


Simplifying
(9b)(9 + -1b) = 32

Remove parenthesis around (9b)
9b(9 + -1b) = 32
(9 * 9b + -1b * 9b) = 32
(81b + -9b2) = 32

Solving
81b + -9b2 = 32

Solving for variable 'b'.

Reorder the terms:
-32 + 81b + -9b2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + 81b + -9b2 = 0

Begin completing the square.  Divide all terms by
-9 the coefficient of the squared term: 

Divide each side by '-9'.
3.555555556 + -9b + b2 = 0

Move the constant term to the right:

Add '-3.555555556' to each side of the equation.
3.555555556 + -9b + -3.555555556 + b2 = 0 + -3.555555556

Reorder the terms:
3.555555556 + -3.555555556 + -9b + b2 = 0 + -3.555555556

Combine like terms: 3.555555556 + -3.555555556 = 0.000000000
0.000000000 + -9b + b2 = 0 + -3.555555556
-9b + b2 = 0 + -3.555555556

Combine like terms: 0 + -3.555555556 = -3.555555556
-9b + b2 = -3.555555556

The b term is -9b.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9b + 20.25 + b2 = -3.555555556 + 20.25

Reorder the terms:
20.25 + -9b + b2 = -3.555555556 + 20.25

Combine like terms: -3.555555556 + 20.25 = 16.694444444
20.25 + -9b + b2 = 16.694444444

Factor a perfect square on the left side:
(b + -4.5)(b + -4.5) = 16.694444444

Calculate the square root of the right side: 4.085883557

Break this problem into two subproblems by setting 
(b + -4.5) equal to 4.085883557 and -4.085883557.

Subproblem 1

b + -4.5 = 4.085883557 Simplifying b + -4.5 = 4.085883557 Reorder the terms: -4.5 + b = 4.085883557 Solving -4.5 + b = 4.085883557 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + b = 4.085883557 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + b = 4.085883557 + 4.5 b = 4.085883557 + 4.5 Combine like terms: 4.085883557 + 4.5 = 8.585883557 b = 8.585883557 Simplifying b = 8.585883557

Subproblem 2

b + -4.5 = -4.085883557 Simplifying b + -4.5 = -4.085883557 Reorder the terms: -4.5 + b = -4.085883557 Solving -4.5 + b = -4.085883557 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + b = -4.085883557 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + b = -4.085883557 + 4.5 b = -4.085883557 + 4.5 Combine like terms: -4.085883557 + 4.5 = 0.414116443 b = 0.414116443 Simplifying b = 0.414116443

Solution

The solution to the problem is based on the solutions from the subproblems. b = {8.585883557, 0.414116443}

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