Solve left(x+yright)^2=x^2+y^2

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

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&y=0\end{matrix}\right.$$

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}.

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

Subtract x^{2} from both sides.

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

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

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

Subtract y^{2} from both sides.

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

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

$$2xy=0$$

The equation is in standard form.

$$2yx=0$$

Divide 0 by 2y.

$$x=0$$

Solve for y (complex solution)

$$\left\{\begin{matrix}\\y=0\text{, }&\text{unconditionally}\\y\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

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}.

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

Subtract y^{2} from both sides.

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

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

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

Subtract x^{2} from both sides.

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

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

$$2xy=0$$

Divide 0 by 2x.

$$y=0$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

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}.

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

Subtract x^{2} from both sides.

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

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

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

Subtract y^{2} from both sides.

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

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

$$2xy=0$$

The equation is in standard form.

$$2yx=0$$

Divide 0 by 2y.

$$x=0$$

Solve for y

$$\left\{\begin{matrix}\\y=0\text{, }&\text{unconditionally}\\y\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

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}.

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

Subtract y^{2} from both sides.

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

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

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

Subtract x^{2} from both sides.

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

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

$$2xy=0$$

Divide 0 by 2x.

$$y=0$$