3z+4=1/2z-8

Solution for 3z+4=1/2z-8 equation:

3z+4=1/2z-8

We move all terms to the left:

3z+4-(1/2z-8)=0


Domain of the equation: 2z-8)!=0

z∈R


We get rid of parentheses

3z-1/2z+8+4=0

We multiply all the terms by the denominator


3z*2z+8*2z+4*2z-1=0

Wy multiply elements

6z^2+16z+8z-1=0

We add all the numbers together, and all the variables

6z^2+24z-1=0

a = 6; b = 24; c = -1;
Δ = b2-4ac
Δ = 242-4·6·(-1)
Δ = 600
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:

$\sqrt{\Delta}=\sqrt{600}=\sqrt{100*6}=\sqrt{100}*\sqrt{6}=10\sqrt{6}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-10\sqrt{6}}{2*6}=\frac{-24-10\sqrt{6}}{12}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+10\sqrt{6}}{2*6}=\frac{-24+10\sqrt{6}}{12}
$