3+1/2y=12-3y

Solution for 3+1/2y=12-3y equation:

3+1/2y=12-3y

We move all terms to the left:

3+1/2y-(12-3y)=0


Domain of the equation: 2y!=0

y!=0/2

y!=0

y∈R


We add all the numbers together, and all the variables

1/2y-(-3y+12)+3=0

We get rid of parentheses

1/2y+3y-12+3=0

We multiply all the terms by the denominator


3y*2y-12*2y+3*2y+1=0

Wy multiply elements

6y^2-24y+6y+1=0

We add all the numbers together, and all the variables

6y^2-18y+1=0

a = 6; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·6·1
Δ = 300
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:

$\sqrt{\Delta}=\sqrt{300}=\sqrt{100*3}=\sqrt{100}*\sqrt{3}=10\sqrt{3}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-10\sqrt{3}}{2*6}=\frac{18-10\sqrt{3}}{12}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+10\sqrt{3}}{2*6}=\frac{18+10\sqrt{3}}{12}
$