b+(b+45)+90+(2b-90)3/2b=540

Solution for b+(b+45)+90+(2b-90)3/2b=540 equation:

b+(b+45)+90+(2b-90)3/2b=540

We move all terms to the left:

b+(b+45)+90+(2b-90)3/2b-(540)=0


Domain of the equation: 2b!=0

b!=0/2

b!=0

b∈R


We add all the numbers together, and all the variables

b+(b+45)+(2b-90)3/2b-450=0

We get rid of parentheses

b+b+(2b-90)3/2b+45-450=0

We multiply all the terms by the denominator


b*2b+b*2b+(2b-90)3+45*2b-450*2b=0

We multiply parentheses

b*2b+b*2b+6b+45*2b-450*2b-270=0

Wy multiply elements

2b^2+2b^2+6b+90b-900b-270=0

We add all the numbers together, and all the variables

4b^2-804b-270=0

a = 4; b = -804; c = -270;
Δ = b2-4ac
Δ = -8042-4·4·(-270)
Δ = 650736
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{650736}=\sqrt{144*4519}=\sqrt{144}*\sqrt{4519}=12\sqrt{4519}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-804)-12\sqrt{4519}}{2*4}=\frac{804-12\sqrt{4519}}{8}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-804)+12\sqrt{4519}}{2*4}=\frac{804+12\sqrt{4519}}{8}
$