-1/2y-1/2y+5/6y=-2/3

Solution for -1/2y-1/2y+5/6y=-2/3 equation:

-1/2y-1/2y+5/6y=-2/3

We move all terms to the left:

-1/2y-1/2y+5/6y-(-2/3)=0


Domain of the equation: 2y!=0

y!=0/2

y!=0

y∈R



Domain of the equation: 6y!=0

y!=0/6

y!=0

y∈R


We get rid of parentheses

-1/2y-1/2y+5/6y+2/3=0

We calculate fractions


144y^2/108y^2+(-54y-1)/108y^2+90y/108y^2=0

We multiply all the terms by the denominator


144y^2+(-54y-1)+90y=0

We add all the numbers together, and all the variables

144y^2+90y+(-54y-1)=0

We get rid of parentheses

144y^2+90y-54y-1=0

We add all the numbers together, and all the variables

144y^2+36y-1=0

a = 144; b = 36; c = -1;
Δ = b2-4ac
Δ = 362-4·144·(-1)
Δ = 1872
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:

$\sqrt{\Delta}=\sqrt{1872}=\sqrt{144*13}=\sqrt{144}*\sqrt{13}=12\sqrt{13}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(36)-12\sqrt{13}}{2*144}=\frac{-36-12\sqrt{13}}{288}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(36)+12\sqrt{13}}{2*144}=\frac{-36+12\sqrt{13}}{288}
$