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## what value of z will result in a zero value

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

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

Thats definitely a cleaner looking answer than what myself and Arthur came up with. I bet this is the correct interpretation.
(-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
[3+√9+180]/10
[3+√189]/10, but 189=9(21)
[3+3√21]/10
(3+3√21)/10 and (3-3√21)/10
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