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

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

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

We move all terms to the left:

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

We multiply parentheses

b+180b+(b+45)+3/2b-8100-540=0

We get rid of parentheses

b+180b+b+3/2b+45-8100-540=0

We multiply all the terms by the denominator


b*2b+180b*2b+b*2b+45*2b-8100*2b-540*2b+3=0

Wy multiply elements

2b^2+360b^2+2b^2+90b-16200b-1080b+3=0

We add all the numbers together, and all the variables

364b^2-17190b+3=0

a = 364; b = -17190; c = +3;
Δ = b2-4ac
Δ = -171902-4·364·3
Δ = 295491732
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:

$\sqrt{\Delta}=\sqrt{295491732}=\sqrt{4*73872933}=\sqrt{4}*\sqrt{73872933}=2\sqrt{73872933}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-17190)-2\sqrt{73872933}}{2*364}=\frac{17190-2\sqrt{73872933}}{728}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-17190)+2\sqrt{73872933}}{2*364}=\frac{17190+2\sqrt{73872933}}{728}
$