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

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

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

We move all terms to the left:

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


Domain of the equation: 2b!=0

b!=0/2

b!=0

b∈R


We multiply parentheses

b+3/2b+(2b-90)+90b+4050-540=0

We get rid of parentheses

b+3/2b+2b+90b-90+4050-540=0

We multiply all the terms by the denominator


b*2b+2b*2b+90b*2b-90*2b+4050*2b-540*2b+3=0

Wy multiply elements

2b^2+4b^2+180b^2-180b+8100b-1080b+3=0

We add all the numbers together, and all the variables

186b^2+6840b+3=0

a = 186; b = 6840; c = +3;
Δ = b2-4ac
Δ = 68402-4·186·3
Δ = 46783368
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:

$\sqrt{\Delta}=\sqrt{46783368}=\sqrt{36*1299538}=\sqrt{36}*\sqrt{1299538}=6\sqrt{1299538}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6840)-6\sqrt{1299538}}{2*186}=\frac{-6840-6\sqrt{1299538}}{372}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6840)+6\sqrt{1299538}}{2*186}=\frac{-6840+6\sqrt{1299538}}{372}
$