15y+6-2/3y=32+0,5y

Solution for 15y+6-2/3y=32+0,5y equation:

15y+6-2/3y=32+0.5y

We move all terms to the left:

15y+6-2/3y-(32+0.5y)=0


Domain of the equation: 3y!=0

y!=0/3

y!=0

y∈R


We add all the numbers together, and all the variables

15y-2/3y-(0.5y+32)+6=0

We get rid of parentheses

15y-2/3y-0.5y-32+6=0

We multiply all the terms by the denominator


15y*3y-(0.5y)*3y-32*3y+6*3y-2=0

We add all the numbers together, and all the variables

15y*3y-(+0.5y)*3y-32*3y+6*3y-2=0

We multiply parentheses

-0y^2+15y*3y-32*3y+6*3y-2=0

Wy multiply elements

-0y^2+45y^2-96y+18y-2=0

We add all the numbers together, and all the variables

44y^2-78y-2=0

a = 44; b = -78; c = -2;
Δ = b2-4ac
Δ = -782-4·44·(-2)
Δ = 6436
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{6436}=\sqrt{4*1609}=\sqrt{4}*\sqrt{1609}=2\sqrt{1609}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-78)-2\sqrt{1609}}{2*44}=\frac{78-2\sqrt{1609}}{88}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-78)+2\sqrt{1609}}{2*44}=\frac{78+2\sqrt{1609}}{88}
$