Solve 0=x^2+zx+1

$$0= { x }^{ 2 } +zx+1$$

Solve for z

$$z=-x-\frac{1}{x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+zx+1=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$zx+1=-x^{2}$$

Subtract 1 from both sides.

$$zx=-x^{2}-1$$

The equation is in standard form.

$$xz=-x^{2}-1$$

Divide both sides by x.

$$\frac{xz}{x}=\frac{-x^{2}-1}{x}$$

Dividing by x undoes the multiplication by x.

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

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

$$z=-x-\frac{1}{x}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{z^{2}-4}-z}{2}$$

$$x=\frac{-\sqrt{z^{2}-4}-z}{2}$$

Solve for x

$$x=\frac{\sqrt{z^{2}-4}-z}{2}$$

$$x=\frac{-\sqrt{z^{2}-4}-z}{2}\text{, }|z|\geq 2$$