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

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

Solve for x (complex solution)

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

Solution Steps

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

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

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

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

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

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

Combine -2xy and 2xy to get 0.

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

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

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

Use the distributive property to multiply 2 by x^{2}+y^{2}.

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

Subtract 2x^{2} from both sides.

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

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

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

Cancel out 2 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

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

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

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

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

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

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

Combine -2xy and 2xy to get 0.

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

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

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

Use the distributive property to multiply 2 by x^{2}+y^{2}.

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

Subtract 2y^{2} from both sides.

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

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

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

Cancel out 2 on both sides.

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

Reorder the terms.

$$\text{true}$$

This is true for any y.

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

Solve for x

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

Solution Steps

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

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

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

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

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

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

Combine -2xy and 2xy to get 0.

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

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

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

Use the distributive property to multiply 2 by x^{2}+y^{2}.

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

Subtract 2x^{2} from both sides.

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

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

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

Cancel out 2 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

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

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

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

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

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

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

Combine -2xy and 2xy to get 0.

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

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

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

Use the distributive property to multiply 2 by x^{2}+y^{2}.

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

Subtract 2y^{2} from both sides.

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

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

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

Cancel out 2 on both sides.

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

Reorder the terms.

$$\text{true}$$

This is true for any y.

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