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

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

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

We move all terms to the left:

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

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

4b^2+4b^2+2b^2-180b+90b-900b+3=0

We add all the numbers together, and all the variables

10b^2-990b+3=0

a = 10; b = -990; c = +3;
Δ = b2-4ac
Δ = -9902-4·10·3
Δ = 979980
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:

$\sqrt{\Delta}=\sqrt{979980}=\sqrt{4*244995}=\sqrt{4}*\sqrt{244995}=2\sqrt{244995}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-990)-2\sqrt{244995}}{2*10}=\frac{990-2\sqrt{244995}}{20}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-990)+2\sqrt{244995}}{2*10}=\frac{990+2\sqrt{244995}}{20}
$