Solve sqrt{2}+sqrt{3}+sqrt{6}=sqrt{x}+sqrt{y}+sqrt{z}

$$\sqrt { 2 } + \sqrt { 3 } + \sqrt { 6 } = \sqrt { x } + \sqrt { y } + \sqrt { z }$$

Solve for x

$$x=\left(-\sqrt{y}-\sqrt{z}+\sqrt{2}+\sqrt{3}+\sqrt{6}\right)^{2}$$

$$y\geq 0\text{ and }z\geq 0\text{ and }-\sqrt{y}-\sqrt{z}+\sqrt{2}+\sqrt{3}+\sqrt{6}\geq 0$$

Steps for Solving Radical Equations

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{2}+\sqrt{3}+\sqrt{6}$$

Subtract \sqrt{y} from both sides.

$$\sqrt{x}+\sqrt{z}=\sqrt{2}+\sqrt{3}+\sqrt{6}-\sqrt{y}$$

Subtract \sqrt{z} from both sides.

$$\sqrt{x}=\sqrt{2}+\sqrt{3}+\sqrt{6}-\sqrt{y}-\sqrt{z}$$

Square both sides of the equation.

$$x=\left(-\sqrt{y}-\sqrt{z}+\sqrt{2}+\sqrt{3}+\sqrt{6}\right)^{2}$$

Solve for y

$$y=\left(-\sqrt{x}-\sqrt{z}+\sqrt{2}+\sqrt{3}+\sqrt{6}\right)^{2}$$

$$x\geq 0\text{ and }z\geq 0\text{ and }-\sqrt{x}-\sqrt{z}+\sqrt{2}+\sqrt{3}+\sqrt{6}\geq 0$$

Steps for Solving Radical Equations

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{2}+\sqrt{3}+\sqrt{6}$$

Subtract \sqrt{x} from both sides.

$$\sqrt{y}+\sqrt{z}=\sqrt{2}+\sqrt{3}+\sqrt{6}-\sqrt{x}$$

Subtract \sqrt{z} from both sides.

$$\sqrt{y}=\sqrt{2}+\sqrt{3}+\sqrt{6}-\sqrt{x}-\sqrt{z}$$

Square both sides of the equation.

$$y=\left(-\sqrt{x}-\sqrt{z}+\sqrt{2}+\sqrt{3}+\sqrt{6}\right)^{2}$$