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

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

Solve for J (complex solution)

$$\left\{\begin{matrix}J=\frac{y^{2}+x-2xy}{\left(x-y\right)^{2}}\text{, }&x\neq y\\J\in \mathrm{C}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(y=1\text{ and }x=1\right)\end{matrix}\right.$$

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}.

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

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

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

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

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

Subtract x from both sides.

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

Subtract y^{2} from both sides.

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

Combine all terms containing J.

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

Divide both sides by -x^{2}+2xy-y^{2}.

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

Dividing by -x^{2}+2xy-y^{2} undoes the multiplication by -x^{2}+2xy-y^{2}.

$$J=\frac{2xy-x-y^{2}}{-x^{2}+2xy-y^{2}}$$

Divide -x-y^{2}+2xy by -x^{2}+2xy-y^{2}.

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

Solve for J

$$\left\{\begin{matrix}J=\frac{y^{2}+x-2xy}{\left(x-y\right)^{2}}\text{, }&x\neq y\\J\in \mathrm{R}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(y=1\text{ and }x=1\right)\end{matrix}\right.$$

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}.

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

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

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

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

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

Subtract x from both sides.

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

Subtract y^{2} from both sides.

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

Combine all terms containing J.

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

Divide both sides by -x^{2}+2xy-y^{2}.

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

Dividing by -x^{2}+2xy-y^{2} undoes the multiplication by -x^{2}+2xy-y^{2}.

$$J=\frac{2xy-x-y^{2}}{-x^{2}+2xy-y^{2}}$$

Divide 2xy-x-y^{2} by -x^{2}+2xy-y^{2}.

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{-2Jy+\sqrt{1-4y+4Jy+4y^{2}-4Jy^{2}}+2y-1}{2J}\text{; }x=-\frac{-2Jy-\sqrt{1-4y+4Jy+4y^{2}-4Jy^{2}}+2y-1}{2J}\text{, }&J\neq 0\\x=-\frac{y^{2}}{1-2y}\text{, }&J=0\text{ and }y\neq \frac{1}{2}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{-2Jy+\sqrt{1-4y+4Jy+4y^{2}-4Jy^{2}}+2y-1}{2J}\text{; }x=-\frac{-2Jy-\sqrt{1-4y+4Jy+4y^{2}-4Jy^{2}}+2y-1}{2J}\text{, }&\left(y\geq \frac{\sqrt{J^{2}-J}-J+1}{2\left(1-J\right)}\text{ or }J\leq 1\right)\text{ and }\left(y\leq -\frac{\sqrt{J^{2}-J}+J-1}{2\left(1-J\right)}\text{ or }J\leq 1\right)\text{ and }\left(J>0\text{ or }y\leq -\frac{\sqrt{J^{2}-J}+J-1}{2\left(1-J\right)}\text{ or }y\geq \frac{\sqrt{J^{2}-J}-J+1}{2\left(1-J\right)}\right)\text{ and }J\neq 0\\x=-\frac{y^{2}}{1-2y}\text{, }&J=0\text{ and }y\neq \frac{1}{2}\end{matrix}\right.$$