z-1/3z=12

Solution for z-1/3z=12 equation:

z-1/3z=12

We move all terms to the left:

z-1/3z-(12)=0


Domain of the equation: 3z!=0

z!=0/3

z!=0

z∈R


We multiply all the terms by the denominator


z*3z-12*3z-1=0

Wy multiply elements

3z^2-36z-1=0

a = 3; b = -36; c = -1;
Δ = b2-4ac
Δ = -362-4·3·(-1)
Δ = 1308
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{1308}=\sqrt{4*327}=\sqrt{4}*\sqrt{327}=2\sqrt{327}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-2\sqrt{327}}{2*3}=\frac{36-2\sqrt{327}}{6}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+2\sqrt{327}}{2*3}=\frac{36+2\sqrt{327}}{6}
$