b+(b+45)+90+(2b-90)+3/2b=240

Solution for b+(b+45)+90+(2b-90)+3/2b=240 equation:

b+(b+45)+90+(2b-90)+3/2b=240

We move all terms to the left:

b+(b+45)+90+(2b-90)+3/2b-(240)=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-90)+3/2b-150=0

We get rid of parentheses

b+b+2b+3/2b+45-90-150=0

We multiply all the terms by the denominator


b*2b+b*2b+2b*2b+45*2b-90*2b-150*2b+3=0

Wy multiply elements

2b^2+2b^2+4b^2+90b-180b-300b+3=0

We add all the numbers together, and all the variables

8b^2-390b+3=0

a = 8; b = -390; c = +3;
Δ = b2-4ac
Δ = -3902-4·8·3
Δ = 152004
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:

$\sqrt{\Delta}=\sqrt{152004}=\sqrt{4*38001}=\sqrt{4}*\sqrt{38001}=2\sqrt{38001}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-390)-2\sqrt{38001}}{2*8}=\frac{390-2\sqrt{38001}}{16}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-390)+2\sqrt{38001}}{2*8}=\frac{390+2\sqrt{38001}}{16}
$