Solve (x-1)^2+(y-3)=5

$$( x - 1 ) ^ { 2 } + ( y - 3 ) = 5$$

Solve for y

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

Solution Steps

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

$$x^{2}-2x+1+y-3=5$$

Subtract 3 from 1 to get -2.

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

Subtract x^{2} from both sides.

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

Add 2x to both sides.

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

Add 2 to both sides.

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

Add 5 and 2 to get 7.

$$y=7-x^{2}+2x$$

Solve for x (complex solution)

$$x=-\sqrt{8-y}+1$$

$$x=\sqrt{8-y}+1$$

Solve for x

$$x=-\sqrt{8-y}+1$$

$$x=\sqrt{8-y}+1\text{, }y\leq 8$$