Solve {left(x-sqrt{8}right)}^2+sqrt{y-5}+left|z-3sqrt{2}right|=0

$${ \left(x- \sqrt{ 8 } \right) }^{ 2 } + \sqrt{ y-5 } + \left| z-3 \sqrt{ 2 } \right| =0$$

Solve for y

$$\left\{\begin{matrix}y=\left(\left(x-2\sqrt{2}\right)^{2}+z-3\sqrt{2}\right)^{2}+5\text{, }&-\left(\left(x-2\sqrt{2}\right)^{2}+z-3\sqrt{2}\right)\geq 0\text{ and }z\geq 3\sqrt{2}\\y=\left(-\left(x-2\sqrt{2}\right)^{2}+z-3\sqrt{2}\right)^{2}+5\text{, }&-\left(x-2\sqrt{2}\right)^{2}+z-3\sqrt{2}\geq 0\text{ and }z\leq 3\sqrt{2}\end{matrix}\right.$$

Steps for Solving Radical Equations

Subtract \left(x-\sqrt{8}\right)^{2}+|z-3\sqrt{2}| from both sides of the equation.

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

Subtracting \left(x-\sqrt{8}\right)^{2}+|z-3\sqrt{2}| from itself leaves 0.

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

Subtract \left(x-\sqrt{8}\right)^{2}+|z-3\sqrt{2}| from 0.

$$\sqrt{y-5}=-\left(\left(x-2\sqrt{2}\right)^{2}+|z-3\sqrt{2}|\right)$$

Square both sides of the equation.

$$y-5=\left(\left(x-2\sqrt{2}\right)^{2}+|z-3\sqrt{2}|\right)^{2}$$

Add 5 to both sides of the equation.

$$y-5-\left(-5\right)=\left(\left(x-2\sqrt{2}\right)^{2}+|z-3\sqrt{2}|\right)^{2}-\left(-5\right)$$

Subtracting -5 from itself leaves 0.

$$y=\left(\left(x-2\sqrt{2}\right)^{2}+|z-3\sqrt{2}|\right)^{2}-\left(-5\right)$$

Subtract -5 from \left(\left(x-2\sqrt{2}\right)^{2}+|z-3\sqrt{2}|\right)^{2}.

$$y=\left(\left(x-2\sqrt{2}\right)^{2}+|z-3\sqrt{2}|\right)^{2}+5$$