10=b(2b-3)

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Solution for 10=b(2b-3) equation:


Simplifying
10 = b(2b + -3)

Reorder the terms:
10 = b(-3 + 2b)
10 = (-3 * b + 2b * b)
10 = (-3b + 2b2)

Solving
10 = -3b + 2b2

Solving for variable 'b'.

Reorder the terms:
10 + 3b + -2b2 = -3b + 3b + 2b2 + -2b2

Combine like terms: -3b + 3b = 0
10 + 3b + -2b2 = 0 + 2b2 + -2b2
10 + 3b + -2b2 = 2b2 + -2b2

Combine like terms: 2b2 + -2b2 = 0
10 + 3b + -2b2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-5 + -1.5b + b2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1.5b + 5 + b2 = 0 + 5

Reorder the terms:
-5 + 5 + -1.5b + b2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1.5b + b2 = 0 + 5
-1.5b + b2 = 0 + 5

Combine like terms: 0 + 5 = 5
-1.5b + b2 = 5

The b term is -1.5b.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5b + 0.5625 + b2 = 5 + 0.5625

Reorder the terms:
0.5625 + -1.5b + b2 = 5 + 0.5625

Combine like terms: 5 + 0.5625 = 5.5625
0.5625 + -1.5b + b2 = 5.5625

Factor a perfect square on the left side:
(b + -0.75)(b + -0.75) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(b + -0.75) equal to 2.358495283 and -2.358495283.

Subproblem 1

b + -0.75 = 2.358495283 Simplifying b + -0.75 = 2.358495283 Reorder the terms: -0.75 + b = 2.358495283 Solving -0.75 + b = 2.358495283 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = 2.358495283 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = 2.358495283 + 0.75 b = 2.358495283 + 0.75 Combine like terms: 2.358495283 + 0.75 = 3.108495283 b = 3.108495283 Simplifying b = 3.108495283

Subproblem 2

b + -0.75 = -2.358495283 Simplifying b + -0.75 = -2.358495283 Reorder the terms: -0.75 + b = -2.358495283 Solving -0.75 + b = -2.358495283 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = -2.358495283 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = -2.358495283 + 0.75 b = -2.358495283 + 0.75 Combine like terms: -2.358495283 + 0.75 = -1.608495283 b = -1.608495283 Simplifying b = -1.608495283

Solution

The solution to the problem is based on the solutions from the subproblems. b = {3.108495283, -1.608495283}

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