(b+45)+(2b-9)+90+3/2b+b=540

Solution for (b+45)+(2b-9)+90+3/2b+b=540 equation:

(b+45)+(2b-9)+90+3/2b+b=540

We move all terms to the left:

(b+45)+(2b-9)+90+3/2b+b-(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+(b+45)+(2b-9)+3/2b-450=0

We get rid of parentheses

b+b+2b+3/2b+45-9-450=0

We multiply all the terms by the denominator


b*2b+b*2b+2b*2b+45*2b-9*2b-450*2b+3=0

Wy multiply elements

2b^2+2b^2+4b^2+90b-18b-900b+3=0

We add all the numbers together, and all the variables

8b^2-828b+3=0

a = 8; b = -828; c = +3;
Δ = b2-4ac
Δ = -8282-4·8·3
Δ = 685488
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:

$\sqrt{\Delta}=\sqrt{685488}=\sqrt{16*42843}=\sqrt{16}*\sqrt{42843}=4\sqrt{42843}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-828)-4\sqrt{42843}}{2*8}=\frac{828-4\sqrt{42843}}{16}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-828)+4\sqrt{42843}}{2*8}=\frac{828+4\sqrt{42843}}{16}
$