Start with
[tex]\dfrac{8}{z}-z=4[/tex]
Note that this implies that z cannot be zero, otherwise in the term 8/z you would have a division by zero.
So, supposing that z is not zero, we multiply both sides by z:
[tex]8-z^2=4z[/tex]
Rearrange the equation into
[tex]z^2+4z-8=0[/tex]
And solve the quadratic equation as usual to get
[tex]z = \dfrac{-4\pm\sqrt{16+32}}{2} = \dfrac{-4\pm\sqrt{48}}{2}= \dfrac{-4\pm4\sqrt{3}}{2} = -2\pm2\sqrt{3}[/tex]
Since both of these solutions are non-zero, we can accept them.
Step-by-step explanation:
8-z^2=4z
8-z^2-4z=0
-z^2-4z+8=0
z^2+4z-8=0
z(z+4)=8
{{z==-5.46410161514},{z==1.46410161514}}
