Solve mz^2+z+1=0

$$m z ^ { 2 } + z + 1 = 0$$

Solve for m

$$m=-\frac{z+1}{z^{2}}$$

$$z\neq 0$$

Steps for Solving Linear Equation

Subtract z from both sides. Anything subtracted from zero gives its negation.

$$mz^{2}+1=-z$$

Subtract 1 from both sides.

$$mz^{2}=-z-1$$

The equation is in standard form.

$$z^{2}m=-z-1$$

Divide both sides by z^{2}.

$$\frac{z^{2}m}{z^{2}}=\frac{-z-1}{z^{2}}$$

Dividing by z^{2} undoes the multiplication by z^{2}.

$$m=\frac{-z-1}{z^{2}}$$

Divide -z-1 by z^{2}.

$$m=-\frac{z+1}{z^{2}}$$

Solve for z (complex solution)

$$\left\{\begin{matrix}z=\frac{\sqrt{1-4m}-1}{2m}\text{; }z=-\frac{\sqrt{1-4m}+1}{2m}\text{, }&m\neq 0\\z=-1\text{, }&m=0\end{matrix}\right.$$

Solve for z

$$\left\{\begin{matrix}z=\frac{\sqrt{1-4m}-1}{2m}\text{; }z=-\frac{\sqrt{1-4m}+1}{2m}\text{, }&m\neq 0\text{ and }m\leq \frac{1}{4}\\z=-1\text{, }&m=0\end{matrix}\right.$$