1/2z-1/3z=1/2+2

Solution for 1/2z-1/3z=1/2+2 equation:

1/2z-1/3z=1/2+2

We move all terms to the left:

1/2z-1/3z-(1/2+2)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R



Domain of the equation: 3z!=0

z!=0/3

z!=0

z∈R


We get rid of parentheses

1/2z-1/3z-2-1/2=0

We calculate fractions


3z/24z^2+(-8z)/24z^2+(-3z)/24z^2-2=0

We multiply all the terms by the denominator


3z+(-8z)+(-3z)-2*24z^2=0

Wy multiply elements

-48z^2+3z+(-8z)+(-3z)=0

We get rid of parentheses

-48z^2+3z-8z-3z=0

We add all the numbers together, and all the variables

-48z^2-8z=0

a = -48; b = -8; c = 0;
Δ = b2-4ac
Δ = -82-4·(-48)·0
Δ = 64
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{64}=8$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-8}{2*-48}=\frac{0}{-96}
=0
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+8}{2*-48}=\frac{16}{-96}
=-1/6
$