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

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

Solve for x (complex solution)

$$x\neq 0$$

$$\left(y=-2\text{ and }|\frac{arg(x^{2})}{2}-arg(x)|<\pi \text{ and }x\neq 0\right)\text{ or }\left(y=2\text{ and }|\frac{arg(x^{2})}{2}-arg(-x)|<\pi \text{ and }x\neq 0\right)$$

Solution Steps

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

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

Add 2\sqrt{x^{2}} to both sides.

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

Subtract yx from both sides of the equation.

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

Reorder the terms.

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

Square both sides of the equation.

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

Expand \left(2\sqrt{x^{2}}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x^{2}}\right)^{2}=\left(-xy\right)^{2}$$

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

$$4x^{2}=\left(-xy\right)^{2}$$

Expand \left(-xy\right)^{2}.

$$4x^{2}=\left(-1\right)^{2}x^{2}y^{2}$$

Calculate -1 to the power of 2 and get 1.

$$4x^{2}=1x^{2}y^{2}$$

Subtract 1x^{2}y^{2} from both sides.

$$4x^{2}-1x^{2}y^{2}=0$$

Reorder the terms.

$$4x^{2}-x^{2}y^{2}=0$$

Reorder the terms.

$$-x^{2}y^{2}+4x^{2}=0$$

Combine all terms containing x.

$$\left(-y^{2}+4\right)x^{2}=0$$

Dividing by -y^{2}+4 undoes the multiplication by -y^{2}+4.

$$x^{2}=\frac{0}{4-y^{2}}$$

Divide 0 by -y^{2}+4.

$$x^{2}=0$$

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{-2\sqrt{x^{2}}}{x}. The expression is undefined.

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

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

$$x\in \emptyset$$

Solve for y (complex solution)

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

$$x\neq 0$$

Solve for x

$$\left\{\begin{matrix}x>0\text{, }&y=-2\\x<0\text{, }&y=2\end{matrix}\right.$$

Solve for y

$$y=-\frac{2|x|}{x}$$

$$x\neq 0$$