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

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

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

We move all terms to the left:

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

8b^2-720b+3=0

a = 8; b = -720; c = +3;
Δ = b2-4ac
Δ = -7202-4·8·3
Δ = 518304
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:

$\sqrt{\Delta}=\sqrt{518304}=\sqrt{16*32394}=\sqrt{16}*\sqrt{32394}=4\sqrt{32394}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-720)-4\sqrt{32394}}{2*8}=\frac{720-4\sqrt{32394}}{16}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-720)+4\sqrt{32394}}{2*8}=\frac{720+4\sqrt{32394}}{16}
$