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

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

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

We move all terms to the left:

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

2b^2+4b^2+2b^2+180b+90b-900b+3=0

We add all the numbers together, and all the variables

8b^2-630b+3=0

a = 8; b = -630; c = +3;
Δ = b2-4ac
Δ = -6302-4·8·3
Δ = 396804
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:

$\sqrt{\Delta}=\sqrt{396804}=\sqrt{4*99201}=\sqrt{4}*\sqrt{99201}=2\sqrt{99201}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-630)-2\sqrt{99201}}{2*8}=\frac{630-2\sqrt{99201}}{16}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-630)+2\sqrt{99201}}{2*8}=\frac{630+2\sqrt{99201}}{16}
$