b(b+45)+90(2b-90)+1.5b=540

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Solution for b(b+45)+90(2b-90)+1.5b=540 equation:


Simplifying
b(b + 45) + 90(2b + -90) + 1.5b = 540

Reorder the terms:
b(45 + b) + 90(2b + -90) + 1.5b = 540
(45 * b + b * b) + 90(2b + -90) + 1.5b = 540
(45b + b2) + 90(2b + -90) + 1.5b = 540

Reorder the terms:
45b + b2 + 90(-90 + 2b) + 1.5b = 540
45b + b2 + (-90 * 90 + 2b * 90) + 1.5b = 540
45b + b2 + (-8100 + 180b) + 1.5b = 540

Reorder the terms:
-8100 + 45b + 180b + 1.5b + b2 = 540

Combine like terms: 45b + 180b = 225b
-8100 + 225b + 1.5b + b2 = 540

Combine like terms: 225b + 1.5b = 226.5b
-8100 + 226.5b + b2 = 540

Solving
-8100 + 226.5b + b2 = 540

Solving for variable 'b'.

Reorder the terms:
-8100 + -540 + 226.5b + b2 = 540 + -540

Combine like terms: -8100 + -540 = -8640
-8640 + 226.5b + b2 = 540 + -540

Combine like terms: 540 + -540 = 0
-8640 + 226.5b + b2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8640' to each side of the equation.
-8640 + 226.5b + 8640 + b2 = 0 + 8640

Reorder the terms:
-8640 + 8640 + 226.5b + b2 = 0 + 8640

Combine like terms: -8640 + 8640 = 0
0 + 226.5b + b2 = 0 + 8640
226.5b + b2 = 0 + 8640

Combine like terms: 0 + 8640 = 8640
226.5b + b2 = 8640

The b term is 226.5b.  Take half its coefficient (113.25).
Square it (12825.5625) and add it to both sides.

Add '12825.5625' to each side of the equation.
226.5b + 12825.5625 + b2 = 8640 + 12825.5625

Reorder the terms:
12825.5625 + 226.5b + b2 = 8640 + 12825.5625

Combine like terms: 8640 + 12825.5625 = 21465.5625
12825.5625 + 226.5b + b2 = 21465.5625

Factor a perfect square on the left side:
(b + 113.25)(b + 113.25) = 21465.5625

Calculate the square root of the right side: 146.511305025

Break this problem into two subproblems by setting 
(b + 113.25) equal to 146.511305025 and -146.511305025.

Subproblem 1

b + 113.25 = 146.511305025 Simplifying b + 113.25 = 146.511305025 Reorder the terms: 113.25 + b = 146.511305025 Solving 113.25 + b = 146.511305025 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-113.25' to each side of the equation. 113.25 + -113.25 + b = 146.511305025 + -113.25 Combine like terms: 113.25 + -113.25 = 0.00 0.00 + b = 146.511305025 + -113.25 b = 146.511305025 + -113.25 Combine like terms: 146.511305025 + -113.25 = 33.261305025 b = 33.261305025 Simplifying b = 33.261305025

Subproblem 2

b + 113.25 = -146.511305025 Simplifying b + 113.25 = -146.511305025 Reorder the terms: 113.25 + b = -146.511305025 Solving 113.25 + b = -146.511305025 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-113.25' to each side of the equation. 113.25 + -113.25 + b = -146.511305025 + -113.25 Combine like terms: 113.25 + -113.25 = 0.00 0.00 + b = -146.511305025 + -113.25 b = -146.511305025 + -113.25 Combine like terms: -146.511305025 + -113.25 = -259.761305025 b = -259.761305025 Simplifying b = -259.761305025

Solution

The solution to the problem is based on the solutions from the subproblems. b = {33.261305025, -259.761305025}

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