z+1/2z=180

Solution for z+1/2z=180 equation:

z+1/2z=180

We move all terms to the left:

z+1/2z-(180)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R


We multiply all the terms by the denominator


z*2z-180*2z+1=0

Wy multiply elements

2z^2-360z+1=0

a = 2; b = -360; c = +1;
Δ = b2-4ac
Δ = -3602-4·2·1
Δ = 129592
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:

$\sqrt{\Delta}=\sqrt{129592}=\sqrt{4*32398}=\sqrt{4}*\sqrt{32398}=2\sqrt{32398}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-360)-2\sqrt{32398}}{2*2}=\frac{360-2\sqrt{32398}}{4}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-360)+2\sqrt{32398}}{2*2}=\frac{360+2\sqrt{32398}}{4}
$