Solve x^2-y^2=(x-y)(x+y)

$${ x }^{ 2 } - { y }^{ 2 } =(x-y)(x+y)$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Subtract x^{2} from both sides.

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

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

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

Cancel out -1 on both sides.

$$y^{2}=y^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

Solution Steps

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Add y^{2} to both sides.

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

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

$$x^{2}=x^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Subtract x^{2} from both sides.

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

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

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

Cancel out -1 on both sides.

$$y^{2}=y^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Solve for y

$$y\in \mathrm{R}$$

Solution Steps

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Add y^{2} to both sides.

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

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

$$x^{2}=x^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$