Solve the equation z^3-z^2-12z=0 . The solutions are z=square , z=square , and z=square

To solve the equation $z^{3}-z^{2}-12z=0$, we can first factor out the common factor of $z$ from each term:<br /><br />$z(z^{2}-z-12)=0$<br /><br />Now, we can set each factor equal to zero and solve for $z$:<br /><br />$z=0$ or $z^{2}-z-12=0$<br /><br />For the quadratic equation $z^{2}-z-12=0$, we can use the quadratic formula to find the solutions:<br /><br />$z=\frac{-(-1)\pm\sqrt{(-1)^{2}-4(1)(-12)}}{2(1)}$<br /><br />$z=\frac{1\pm\sqrt{1+48}}{2}$<br /><br />$z=\frac{1\pm\sqrt{49}}{2}$<br /><br />$z=\frac{1\pm7}{2}$<br /><br />So, the solutions are $z=0$, $z=\frac{8}{2}=4$, and $z=\frac{-6}{2}=-3$.