Solve -(x-1)^2+(y-3)^2=(x-2)^2+(y-1)^2

$$- ( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = ( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 }$$

Solve for y

$$y=-\frac{x^{2}}{2}+\frac{3x}{2}+\frac{3}{4}$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-\left(x^{2}-2x+1\right)+\left(y-3\right)^{2}=\left(x-2\right)^{2}+\left(y-1\right)^{2}$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$-x^{2}+2x-1+\left(y-3\right)^{2}=\left(x-2\right)^{2}+\left(y-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-3\right)^{2}.

$$-x^{2}+2x-1+y^{2}-6y+9=\left(x-2\right)^{2}+\left(y-1\right)^{2}$$

Add -1 and 9 to get 8.

$$-x^{2}+2x+8+y^{2}-6y=\left(x-2\right)^{2}+\left(y-1\right)^{2}$$

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

$$-x^{2}+2x+8+y^{2}-6y=x^{2}-4x+4+\left(y-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-1\right)^{2}.

$$-x^{2}+2x+8+y^{2}-6y=x^{2}-4x+4+y^{2}-2y+1$$

Add 4 and 1 to get 5.

$$-x^{2}+2x+8+y^{2}-6y=x^{2}-4x+5+y^{2}-2y$$

Subtract y^{2} from both sides.

$$-x^{2}+2x+8+y^{2}-6y-y^{2}=x^{2}-4x+5-2y$$

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

$$-x^{2}+2x+8-6y=x^{2}-4x+5-2y$$

Add 2y to both sides.

$$-x^{2}+2x+8-6y+2y=x^{2}-4x+5$$

Combine -6y and 2y to get -4y.

$$-x^{2}+2x+8-4y=x^{2}-4x+5$$

Add x^{2} to both sides.

$$2x+8-4y=x^{2}-4x+5+x^{2}$$

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

$$2x+8-4y=2x^{2}-4x+5$$

Subtract 2x from both sides.

$$8-4y=2x^{2}-4x+5-2x$$

Combine -4x and -2x to get -6x.

$$8-4y=2x^{2}-6x+5$$

Subtract 8 from both sides.

$$-4y=2x^{2}-6x+5-8$$

Subtract 8 from 5 to get -3.

$$-4y=2x^{2}-6x-3$$

Divide both sides by -4.

$$\frac{-4y}{-4}=\frac{2x^{2}-6x-3}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y=\frac{2x^{2}-6x-3}{-4}$$

Divide 2x^{2}-6x-3 by -4.

$$y=-\frac{x^{2}}{2}+\frac{3x}{2}+\frac{3}{4}$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{15-8y}+3}{2}$$

$$x=\frac{\sqrt{15-8y}+3}{2}$$

Solve for x

$$x=\frac{-\sqrt{15-8y}+3}{2}$$

$$x=\frac{\sqrt{15-8y}+3}{2}\text{, }y\leq \frac{15}{8}$$