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

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

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

We move all terms to the left:

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

We get rid of parentheses

b+b+2b+3/4b+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^2-1260b+3=0

a = 16; b = -1260; c = +3;
Δ = b2-4ac
Δ = -12602-4·16·3
Δ = 1587408
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:

$\sqrt{\Delta}=\sqrt{1587408}=\sqrt{16*99213}=\sqrt{16}*\sqrt{99213}=4\sqrt{99213}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1260)-4\sqrt{99213}}{2*16}=\frac{1260-4\sqrt{99213}}{32}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1260)+4\sqrt{99213}}{2*16}=\frac{1260+4\sqrt{99213}}{32}
$