1/2z+4=5/z-8

Solution for 1/2z+4=5/z-8 equation:

1/2z+4=5/z-8

We move all terms to the left:

1/2z+4-(5/z-8)=0


Domain of the equation: 2z!=0

z!=0/2

z!=0

z∈R



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

z∈R


We get rid of parentheses

1/2z-5/z+8+4=0

We calculate fractions


z/2z^2+(-10z)/2z^2+8+4=0

We add all the numbers together, and all the variables

z/2z^2+(-10z)/2z^2+12=0

We multiply all the terms by the denominator


z+(-10z)+12*2z^2=0

Wy multiply elements

24z^2+z+(-10z)=0

We get rid of parentheses

24z^2+z-10z=0

We add all the numbers together, and all the variables

24z^2-9z=0

a = 24; b = -9; c = 0;
Δ = b2-4ac
Δ = -92-4·24·0
Δ = 81
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{81}=9$
$z_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-9}{2*24}=\frac{0}{48}
=0
$
$z_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+9}{2*24}=\frac{18}{48}
=3/8
$