3z=1/2z+22

Solution for 3z=1/2z+22 equation:

3z=1/2z+22

We move all terms to the left:

3z-(1/2z+22)=0


Domain of the equation: 2z+22)!=0

z∈R


We get rid of parentheses

3z-1/2z-22=0

We multiply all the terms by the denominator


3z*2z-22*2z-1=0

Wy multiply elements

6z^2-44z-1=0

a = 6; b = -44; c = -1;
Δ = b2-4ac
Δ = -442-4·6·(-1)
Δ = 1960
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:

$\sqrt{\Delta}=\sqrt{1960}=\sqrt{196*10}=\sqrt{196}*\sqrt{10}=14\sqrt{10}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-44)-14\sqrt{10}}{2*6}=\frac{44-14\sqrt{10}}{12}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-44)+14\sqrt{10}}{2*6}=\frac{44+14\sqrt{10}}{12}
$