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

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

Solve for y (complex solution)

$$\left\{\begin{matrix}\\y=\frac{x}{4}\text{, }&\text{unconditionally}\\y\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

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}.

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

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

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

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

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

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

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

Combine 2xy and 2xy to get 4xy.

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

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

$$4xy=x^{2}$$

Divide both sides by 4x.

$$\frac{4xy}{4x}=\frac{x^{2}}{4x}$$

Dividing by 4x undoes the multiplication by 4x.

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

Divide x^{2} by 4x.

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

Solve for y

$$\left\{\begin{matrix}\\y=\frac{x}{4}\text{, }&\text{unconditionally}\\y\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

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}.

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

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

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

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

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

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

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

Combine 2xy and 2xy to get 4xy.

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

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

$$4xy=x^{2}$$

Divide both sides by 4x.

$$\frac{4xy}{4x}=\frac{x^{2}}{4x}$$

Dividing by 4x undoes the multiplication by 4x.

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

Divide x^{2} by 4x.

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

Solve for x

$$x=4y$$

$$x=0$$