1/2y+6+3/2y=2y-6y-30

Solution for 1/2y+6+3/2y=2y-6y-30 equation:

1/2y+6+3/2y=2y-6y-30

We move all terms to the left:

1/2y+6+3/2y-(2y-6y-30)=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+3/2y-(-4y-30)+6=0

We get rid of parentheses

1/2y+3/2y+4y+30+6=0

We multiply all the terms by the denominator


4y*2y+30*2y+6*2y+1+3=0

We add all the numbers together, and all the variables

4y*2y+30*2y+6*2y+4=0

Wy multiply elements

8y^2+60y+12y+4=0

We add all the numbers together, and all the variables

8y^2+72y+4=0

a = 8; b = 72; c = +4;
Δ = b2-4ac
Δ = 722-4·8·4
Δ = 5056
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:

$\sqrt{\Delta}=\sqrt{5056}=\sqrt{64*79}=\sqrt{64}*\sqrt{79}=8\sqrt{79}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(72)-8\sqrt{79}}{2*8}=\frac{-72-8\sqrt{79}}{16}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(72)+8\sqrt{79}}{2*8}=\frac{-72+8\sqrt{79}}{16}
$