3z-1/2z+4=1

Solution for 3z-1/2z+4=1 equation:

3z-1/2z+4=1

We move all terms to the left:

3z-1/2z+4-(1)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R


We add all the numbers together, and all the variables

3z-1/2z+3=0

We multiply all the terms by the denominator


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

Wy multiply elements

6z^2+6z-1=0

a = 6; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·6·(-1)
Δ = 60
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:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{15}}{2*6}=\frac{-6-2\sqrt{15}}{12}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{15}}{2*6}=\frac{-6+2\sqrt{15}}{12}
$