18=.5b(2b+1)

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Solution for 18=.5b(2b+1) equation:


Simplifying
18 = 0.5b(2b + 1)

Reorder the terms:
18 = 0.5b(1 + 2b)
18 = (1 * 0.5b + 2b * 0.5b)
18 = (0.5b + 1b2)

Solving
18 = 0.5b + 1b2

Solving for variable 'b'.

Reorder the terms:
18 + -0.5b + -1b2 = 0.5b + -0.5b + 1b2 + -1b2

Combine like terms: 0.5b + -0.5b = 0.0
18 + -0.5b + -1b2 = 0.0 + 1b2 + -1b2
18 + -0.5b + -1b2 = 1b2 + -1b2

Combine like terms: 1b2 + -1b2 = 0
18 + -0.5b + -1b2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-18 + 0.5b + b2 = 0

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 0.5b + 18 + b2 = 0 + 18

Reorder the terms:
-18 + 18 + 0.5b + b2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 0.5b + b2 = 0 + 18
0.5b + b2 = 0 + 18

Combine like terms: 0 + 18 = 18
0.5b + b2 = 18

The b term is 0.5b.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5b + 0.0625 + b2 = 18 + 0.0625

Reorder the terms:
0.0625 + 0.5b + b2 = 18 + 0.0625

Combine like terms: 18 + 0.0625 = 18.0625
0.0625 + 0.5b + b2 = 18.0625

Factor a perfect square on the left side:
(b + 0.25)(b + 0.25) = 18.0625

Calculate the square root of the right side: 4.25

Break this problem into two subproblems by setting 
(b + 0.25) equal to 4.25 and -4.25.

Subproblem 1

b + 0.25 = 4.25 Simplifying b + 0.25 = 4.25 Reorder the terms: 0.25 + b = 4.25 Solving 0.25 + b = 4.25 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = 4.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = 4.25 + -0.25 b = 4.25 + -0.25 Combine like terms: 4.25 + -0.25 = 4 b = 4 Simplifying b = 4

Subproblem 2

b + 0.25 = -4.25 Simplifying b + 0.25 = -4.25 Reorder the terms: 0.25 + b = -4.25 Solving 0.25 + b = -4.25 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = -4.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = -4.25 + -0.25 b = -4.25 + -0.25 Combine like terms: -4.25 + -0.25 = -4.5 b = -4.5 Simplifying b = -4.5

Solution

The solution to the problem is based on the solutions from the subproblems. b = {4, -4.5}

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