Solve y=frac{sqrt{-x^2}}{x}

$$y = \frac { \sqrt { - x ^ { 2 } } } { x }$$

Solve for x

$$x\neq 0$$

$$\left(arg(-ix)<\pi \text{ and }x\neq 0\text{ and }y=-i\right)\text{ or }\left(arg(ix)<\pi \text{ and }x\neq 0\text{ and }y=i\right)$$

Solution Steps

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$yx=\sqrt{-x^{2}}$$

Subtract \sqrt{-x^{2}} from both sides.

$$yx-\sqrt{-x^{2}}=0$$

Subtract yx from both sides of the equation.

$$-\sqrt{-x^{2}}=-yx$$

Cancel out -1 on both sides.

$$\sqrt{-x^{2}}=yx$$

Square both sides of the equation.

$$\left(\sqrt{-x^{2}}\right)^{2}=\left(yx\right)^{2}$$

Calculate \sqrt{-x^{2}} to the power of 2 and get -x^{2}.

$$-x^{2}=\left(yx\right)^{2}$$

Expand \left(yx\right)^{2}.

$$-x^{2}=y^{2}x^{2}$$

Subtract y^{2}x^{2} from both sides.

$$-x^{2}-y^{2}x^{2}=0$$

Reorder the terms.

$$-x^{2}y^{2}-x^{2}=0$$

Combine all terms containing x.

$$\left(-y^{2}-1\right)x^{2}=0$$

Dividing by -y^{2}-1 undoes the multiplication by -y^{2}-1.

$$x^{2}=\frac{0}{-y^{2}-1}$$

Divide 0 by -y^{2}-1.

$$x^{2}=0$$

Take the square root of both sides of the equation.

$$x=0 x=0$$

The equation is now solved. Solutions are the same.

$$x=0$$

Substitute 0 for x in the equation y=\frac{\sqrt{-x^{2}}}{x}. The expression is undefined.

$$y=\frac{\sqrt{-0^{2}}}{0}$$

Equation \sqrt{-x^{2}}=xy has no solutions.

$$x\in \emptyset$$

Solve for y

$$y=\frac{\sqrt{-x^{2}}}{x}$$

$$x\neq 0$$