-7z+12=20z+8-1/2z

Solution for -7z+12=20z+8-1/2z equation:

-7z+12=20z+8-1/2z

We move all terms to the left:

-7z+12-(20z+8-1/2z)=0


Domain of the equation: 2z)!=0

z!=0/1

z!=0

z∈R


We add all the numbers together, and all the variables

-7z-(20z-1/2z+8)+12=0

We get rid of parentheses

-7z-20z+1/2z-8+12=0

We multiply all the terms by the denominator


-7z*2z-20z*2z-8*2z+12*2z+1=0

Wy multiply elements

-14z^2-40z^2-16z+24z+1=0

We add all the numbers together, and all the variables

-54z^2+8z+1=0

a = -54; b = 8; c = +1;
Δ = b2-4ac
Δ = 82-4·(-54)·1
Δ = 280
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:

$\sqrt{\Delta}=\sqrt{280}=\sqrt{4*70}=\sqrt{4}*\sqrt{70}=2\sqrt{70}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-2\sqrt{70}}{2*-54}=\frac{-8-2\sqrt{70}}{-108}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+2\sqrt{70}}{2*-54}=\frac{-8+2\sqrt{70}}{-108}
$