1/2z+4=4z-6

Solution for 1/2z+4=4z-6 equation:

1/2z+4=4z-6

We move all terms to the left:

1/2z+4-(4z-6)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R


We get rid of parentheses

1/2z-4z+6+4=0

We multiply all the terms by the denominator


-4z*2z+6*2z+4*2z+1=0

Wy multiply elements

-8z^2+12z+8z+1=0

We add all the numbers together, and all the variables

-8z^2+20z+1=0

a = -8; b = 20; c = +1;
Δ = b2-4ac
Δ = 202-4·(-8)·1
Δ = 432
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:

$\sqrt{\Delta}=\sqrt{432}=\sqrt{144*3}=\sqrt{144}*\sqrt{3}=12\sqrt{3}$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-12\sqrt{3}}{2*-8}=\frac{-20-12\sqrt{3}}{-16}
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+12\sqrt{3}}{2*-8}=\frac{-20+12\sqrt{3}}{-16}
$