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

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

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

We move all terms to the left:

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

8b^2-1170b+3=0

a = 8; b = -1170; c = +3;
Δ = b2-4ac
Δ = -11702-4·8·3
Δ = 1368804
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:

$\sqrt{\Delta}=\sqrt{1368804}=\sqrt{4*342201}=\sqrt{4}*\sqrt{342201}=2\sqrt{342201}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1170)-2\sqrt{342201}}{2*8}=\frac{1170-2\sqrt{342201}}{16}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1170)+2\sqrt{342201}}{2*8}=\frac{1170+2\sqrt{342201}}{16}
$