Solve iz/z-i=z

$$\frac { i z } { z - i } = z$$

Solve for z

$$z=2i$$

$$z=0$$

Steps Using Factoring

Variable z cannot be equal to i since division by zero is not defined. Multiply both sides of the equation by z-i.

$$iz=z\left(z-i\right)$$

Use the distributive property to multiply z by z-i.

$$iz=z^{2}-iz$$

Subtract z^{2} from both sides.

$$iz-z^{2}=-iz$$

Subtract -iz from both sides.

$$iz-z^{2}-\left(-iz\right)=0$$

Combine iz and iz to get 2iz.

$$2iz-z^{2}=0$$

Factor out z.

$$z\left(2i-z\right)=0$$

To find equation solutions, solve z=0 and 2i-z=0.

$$z=0 z=2i$$

Steps Using the Quadratic Formula

Variable z cannot be equal to i since division by zero is not defined. Multiply both sides of the equation by z-i.

$$iz=z\left(z-i\right)$$

Use the distributive property to multiply z by z-i.

$$iz=z^{2}-iz$$

Subtract z^{2} from both sides.

$$iz-z^{2}=-iz$$

Subtract -iz from both sides.

$$iz-z^{2}-\left(-iz\right)=0$$

Combine iz and iz to get 2iz.

$$2iz-z^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-z^{2}+2iz=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2i for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-2i±\sqrt{\left(2i\right)^{2}}}{2\left(-1\right)}$$

Take the square root of \left(2i\right)^{2}.

$$z=\frac{-2i±2i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$z=\frac{-2i±2i}{-2}$$

Now solve the equation z=\frac{-2i±2i}{-2} when ± is plus. Add -2i to 2i.

$$z=\frac{0}{-2}$$

Divide 0 by -2.

$$z=0$$

Now solve the equation z=\frac{-2i±2i}{-2} when ± is minus. Subtract 2i from -2i.

$$z=\frac{-4i}{-2}$$

Divide -4i by -2.

$$z=2i$$

The equation is now solved.

$$z=0 z=2i$$

Steps for Completing the Square

Variable z cannot be equal to i since division by zero is not defined. Multiply both sides of the equation by z-i.

$$iz=z\left(z-i\right)$$

Use the distributive property to multiply z by z-i.

$$iz=z^{2}-iz$$

Subtract z^{2} from both sides.

$$iz-z^{2}=-iz$$

Subtract -iz from both sides.

$$iz-z^{2}-\left(-iz\right)=0$$

Combine iz and iz to get 2iz.

$$2iz-z^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-z^{2}+2iz=0$$

Divide both sides by -1.

$$\frac{-z^{2}+2iz}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$z^{2}+\frac{2i}{-1}z=\frac{0}{-1}$$

Divide 2i by -1.

$$z^{2}-2iz=\frac{0}{-1}$$

Divide 0 by -1.

$$z^{2}-2iz=0$$

Divide -2i, the coefficient of the x term, by 2 to get -i. Then add the square of -i to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-2iz+\left(-i\right)^{2}=\left(-i\right)^{2}$$

Square -i.

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

Factor z^{2}-2iz-1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-i\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-i\right)^{2}}=\sqrt{-1}$$

Simplify.

$$z-i=i z-i=-i$$

Add i to both sides of the equation.

$$z=2i z=0$$