b+(b+45)+3/4b+90+(2b-90)=540

Solution for b+(b+45)+3/4b+90+(2b-90)=540 equation:

b+(b+45)+3/4b+90+(2b-90)=540

We move all terms to the left:

b+(b+45)+3/4b+90+(2b-90)-(540)=0


Domain of the equation: 4b!=0

b!=0/4

b!=0

b∈R


We add all the numbers together, and all the variables

b+(b+45)+3/4b+(2b-90)-450=0

We get rid of parentheses

b+b+3/4b+2b+45-90-450=0

We multiply all the terms by the denominator


b*4b+b*4b+2b*4b+45*4b-90*4b-450*4b+3=0

Wy multiply elements

4b^2+4b^2+8b^2+180b-360b-1800b+3=0

We add all the numbers together, and all the variables

16b^2-1980b+3=0

a = 16; b = -1980; c = +3;
Δ = b2-4ac
Δ = -19802-4·16·3
Δ = 3920208
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:

$\sqrt{\Delta}=\sqrt{3920208}=\sqrt{16*245013}=\sqrt{16}*\sqrt{245013}=4\sqrt{245013}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1980)-4\sqrt{245013}}{2*16}=\frac{1980-4\sqrt{245013}}{32}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1980)+4\sqrt{245013}}{2*16}=\frac{1980+4\sqrt{245013}}{32}
$