2/3b+1/3b+1+b=15

Solution for 2/3b+1/3b+1+b=15 equation:

2/3b+1/3b+1+b=15

We move all terms to the left:

2/3b+1/3b+1+b-(15)=0


Domain of the equation: 3b!=0

b!=0/3

b!=0

b∈R


We add all the numbers together, and all the variables

b+2/3b+1/3b-14=0

We multiply all the terms by the denominator


b*3b-14*3b+2+1=0

We add all the numbers together, and all the variables

b*3b-14*3b+3=0

Wy multiply elements

3b^2-42b+3=0

a = 3; b = -42; c = +3;
Δ = b2-4ac
Δ = -422-4·3·3
Δ = 1728
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:

$\sqrt{\Delta}=\sqrt{1728}=\sqrt{576*3}=\sqrt{576}*\sqrt{3}=24\sqrt{3}$
$b_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-24\sqrt{3}}{2*3}=\frac{42-24\sqrt{3}}{6}
$
$b_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+24\sqrt{3}}{2*3}=\frac{42+24\sqrt{3}}{6}
$