Find the values of the variable for which the following expression is defined

(z-2)/(z^2-5z-24)

z^2 - 5z - 24 <> 0
z^2 - 8z + 3z - 24 <> 0
z(z - 8) + 3(z - 8) <> 0
(z + 3)(z - 8) <> 0
z + 3 <> 0 and z - 8 <> 0
z <> -3 and z <> 8
z can be anything except -3 or 8

If denominator is 0, then the equation is undefined. So, you can just factor the denominator. z^2 -5z -24
= (z-8)(z+3) so for values where z = 8 and z = -3, the equation is undefined

Factorise the bottom:
(z-8)(z+3)
The denominator cannot equal 0, so z doesn't equal 8 or -3