what value of z will result in a zero value

Question

-8/3z-1+9/3z+1=5/9z^2-1

Answer 1

Assuming this is correct interpretation:

-8/(3z-1) + 9/(3z+1) = 5/(9z^2-1)

-8/(3z-1) + 9/(3z+1) = 5/((3z-1)(3z+1))

-8/(3z-1) + 9/(3z+1) = 5/((3z-1)(3z+1))

Multiply both sides by (3z-1)(3z+1):

-8(3z+1) +9(3z-1) = 5

-24z - 8 + 27z - 9 = 5

3z = 22

z = 22/3

Answer 2

(-8/3)z-1+(9/3)z+1=(5/9)z^2-1

simplify the left side

(1/3)z=(5/9)z^2-1

0=(5/9)z^2-(1/3)z-1

multiply both sides by 9

0=5z^2-3z-9

use the quadratic equation

[3+√9+180]/10

[3+√189]/10, but 189=9(21)

[3+3√21]/10

(3+3√21)/10 and (3-3√21)/10

Answer 3

First Simplify and set the equation equal to Zero.
-(5/9)z^2+(1/3)z+1=0
This equation can be solved using the quadratic equation. 
x=(b±√(b²+4ac))/(2a)
a=-5/9
b=1/3
c=1
Plug in the variables to the quadratic equation and solve.
Solve for "x", which is z in this problem. You will get two answers.
z=-1.07477 = -3(√21-1)/10
z=1.67477 = -3(√21+1)/10