Solve (x+4)^2+(y-1)^2+49=sqrt{(x-3)^2}+(y-5)^2+4

$$( x + 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + 49 = \sqrt { ( x - 3 ) ^ { 2 } } + ( y - 5 ) ^ { 2 } + 4$$

Solve for y

$$y=\frac{-\left(x+4\right)^{2}+|x-3|-21}{8}$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{2}+8x+16+\left(y-1\right)^{2}+49=\sqrt{\left(x-3\right)^{2}}+\left(y-5\right)^{2}+4$$

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

$$x^{2}+8x+16+y^{2}-2y+1+49=\sqrt{\left(x-3\right)^{2}}+\left(y-5\right)^{2}+4$$

Add 16 and 1 to get 17.

$$x^{2}+8x+17+y^{2}-2y+49=\sqrt{\left(x-3\right)^{2}}+\left(y-5\right)^{2}+4$$

Add 17 and 49 to get 66.

$$x^{2}+8x+66+y^{2}-2y=\sqrt{\left(x-3\right)^{2}}+\left(y-5\right)^{2}+4$$

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

$$x^{2}+8x+66+y^{2}-2y=\sqrt{x^{2}-6x+9}+\left(y-5\right)^{2}+4$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-5\right)^{2}.

$$x^{2}+8x+66+y^{2}-2y=\sqrt{x^{2}-6x+9}+y^{2}-10y+25+4$$

Add 25 and 4 to get 29.

$$x^{2}+8x+66+y^{2}-2y=\sqrt{x^{2}-6x+9}+y^{2}-10y+29$$

Subtract y^{2} from both sides.

$$x^{2}+8x+66+y^{2}-2y-y^{2}=\sqrt{x^{2}-6x+9}-10y+29$$

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

$$x^{2}+8x+66-2y=\sqrt{x^{2}-6x+9}-10y+29$$

Add 10y to both sides.

$$x^{2}+8x+66-2y+10y=\sqrt{x^{2}-6x+9}+29$$

Combine -2y and 10y to get 8y.

$$x^{2}+8x+66+8y=\sqrt{x^{2}-6x+9}+29$$

Subtract x^{2} from both sides.

$$8x+66+8y=\sqrt{x^{2}-6x+9}+29-x^{2}$$

Subtract 8x from both sides.

$$66+8y=\sqrt{x^{2}-6x+9}+29-x^{2}-8x$$

Subtract 66 from both sides.

$$8y=\sqrt{x^{2}-6x+9}+29-x^{2}-8x-66$$

Subtract 66 from 29 to get -37.

$$8y=\sqrt{x^{2}-6x+9}-37-x^{2}-8x$$

The equation is in standard form.

$$8y=-x^{2}+\sqrt{x^{2}-6x+9}-8x-37$$

Divide both sides by 8.

$$\frac{8y}{8}=\frac{-x^{2}+|x-3|-8x-37}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$y=\frac{-x^{2}+|x-3|-8x-37}{8}$$

Divide |x-3|-37-x^{2}-8x by 8.

$$y=-\frac{x^{2}}{8}+\frac{|x-3|}{8}-x-\frac{37}{8}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{-32y-111}-7}{2}\text{, }&y\leq -\frac{35}{4}\\x=\frac{-\sqrt{-32y-55}-9}{2}\text{, }&y\leq -\frac{55}{32}\\x=\frac{\sqrt{-32y-55}-9}{2}\text{, }&y\geq -\frac{35}{4}\text{ and }y\leq -\frac{55}{32}\end{matrix}\right.$$