-1/3z-6=2z+1

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

-1/3z-6=2z+1

We move all terms to the left:

-1/3z-6-(2z+1)=0


Domain of the equation: 3z!=0

z!=0/3

z!=0

z∈R


We get rid of parentheses

-1/3z-2z-1-6=0

We multiply all the terms by the denominator


-2z*3z-1*3z-6*3z-1=0

Wy multiply elements

-6z^2-3z-18z-1=0

We add all the numbers together, and all the variables

-6z^2-21z-1=0

a = -6; b = -21; c = -1;
Δ = b2-4ac
Δ = -212-4·(-6)·(-1)
Δ = 417
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-21)-\sqrt{417}}{2*-6}=\frac{21-\sqrt{417}}{-12}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-21)+\sqrt{417}}{2*-6}=\frac{21+\sqrt{417}}{-12}
$