Solve left(x-yright)^2-left(x+yright)^2=-12

$${ \left(x-y \right) }^{ 2 } - { \left(x+y \right) }^{ 2 } = -12$$

Solve for x

$$x=\frac{3}{y}$$

$$y\neq 0$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$x^{2}-2xy+y^{2}-\left(x+y\right)^{2}=-12$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$x^{2}-2xy+y^{2}-\left(x^{2}+2xy+y^{2}\right)=-12$$

To find the opposite of x^{2}+2xy+y^{2}, find the opposite of each term.

$$x^{2}-2xy+y^{2}-x^{2}-2xy-y^{2}=-12$$

Combine x^{2} and -x^{2} to get 0.

$$-2xy+y^{2}-2xy-y^{2}=-12$$

Combine -2xy and -2xy to get -4xy.

$$-4xy+y^{2}-y^{2}=-12$$

Combine y^{2} and -y^{2} to get 0.

$$-4xy=-12$$

The equation is in standard form.

$$\left(-4y\right)x=-12$$

Divide both sides by -4y.

$$\frac{\left(-4y\right)x}{-4y}=-\frac{12}{-4y}$$

Dividing by -4y undoes the multiplication by -4y.

$$x=-\frac{12}{-4y}$$

Divide -12 by -4y.

$$x=\frac{3}{y}$$

Solve for y

$$y=\frac{3}{x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$x^{2}-2xy+y^{2}-\left(x+y\right)^{2}=-12$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$x^{2}-2xy+y^{2}-\left(x^{2}+2xy+y^{2}\right)=-12$$

To find the opposite of x^{2}+2xy+y^{2}, find the opposite of each term.

$$x^{2}-2xy+y^{2}-x^{2}-2xy-y^{2}=-12$$

Combine x^{2} and -x^{2} to get 0.

$$-2xy+y^{2}-2xy-y^{2}=-12$$

Combine -2xy and -2xy to get -4xy.

$$-4xy+y^{2}-y^{2}=-12$$

Combine y^{2} and -y^{2} to get 0.

$$-4xy=-12$$

The equation is in standard form.

$$\left(-4x\right)y=-12$$

Divide both sides by -4x.

$$\frac{\left(-4x\right)y}{-4x}=-\frac{12}{-4x}$$

Dividing by -4x undoes the multiplication by -4x.

$$y=-\frac{12}{-4x}$$

Divide -12 by -4x.

$$y=\frac{3}{x}$$