Question

Solve the following equation for $z.$
$\frac{4}{z-1}-\frac{5}{z+2}=\frac{3}{z},z\ne 1,0,-2$

Answer 1

$\frac{4}{z-1}-\frac{5}{z+2}=\frac{3}{z}$

$\frac{4z+8-5z+5}{(z-1)(z+2)}=\frac{3}{z}$

$⟹(13-z)z=3(z-1)(z+2)$

$⟹13z-z^2=3z^2+3z-6⟹4z^2-10z-6=0⟹z=\frac{10\pm \sqrt{100+96}}{8}=\frac{10\pm 14}{8}=3,-\frac{1}{2}$