3/2b+b+(b+45)+(2b-90)+90=180

Solution for 3/2b+b+(b+45)+(2b-90)+90=180 equation:

3/2b+b+(b+45)+(2b-90)+90=180

We move all terms to the left:

3/2b+b+(b+45)+(2b-90)+90-(180)=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+3/2b+(b+45)+(2b-90)-90=0

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

8b^2-270b+3=0

a = 8; b = -270; c = +3;
Δ = b2-4ac
Δ = -2702-4·8·3
Δ = 72804
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:

$\sqrt{\Delta}=\sqrt{72804}=\sqrt{4*18201}=\sqrt{4}*\sqrt{18201}=2\sqrt{18201}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-270)-2\sqrt{18201}}{2*8}=\frac{270-2\sqrt{18201}}{16}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-270)+2\sqrt{18201}}{2*8}=\frac{270+2\sqrt{18201}}{16}
$