Solve 43-12sqrt{7}=(sqrt{x}-sqrt{y})

$$43 - 12 \sqrt { 7 } = ( \sqrt { x } - \sqrt { y } )$$

Solve for x

$$x=\left(\sqrt{y}+43-12\sqrt{7}\right)^{2}$$

$$y\geq 0$$

Steps for Solving Radical Equations

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

$$\sqrt{x}-\sqrt{y}=43-12\sqrt{7}$$

Add \sqrt{y} to both sides.

$$\sqrt{x}=43-12\sqrt{7}+\sqrt{y}$$

Square both sides of the equation.

$$x=\left(\sqrt{y}+43-12\sqrt{7}\right)^{2}$$

Solve for y

$$y=\left(-\sqrt{x}+43-12\sqrt{7}\right)^{2}$$

$$x\geq 0\text{ and }-\left(-\sqrt{x}+43-12\sqrt{7}\right)\geq 0$$

Steps for Solving Radical Equations

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

$$\sqrt{x}-\sqrt{y}=43-12\sqrt{7}$$

Subtract \sqrt{x} from both sides.

$$-\sqrt{y}=43-12\sqrt{7}-\sqrt{x}$$

Divide both sides by -1.

$$\frac{-\sqrt{y}}{-1}=\frac{-\sqrt{x}+43-12\sqrt{7}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$\sqrt{y}=\frac{-\sqrt{x}+43-12\sqrt{7}}{-1}$$

Divide 43-12\sqrt{7}-\sqrt{x} by -1.

$$\sqrt{y}=\sqrt{x}+12\sqrt{7}-43$$

Square both sides of the equation.

$$y=\left(\sqrt{x}+12\sqrt{7}-43\right)^{2}$$

Solve for x (complex solution)

$$x=\left(\sqrt{y}+43-12\sqrt{7}\right)^{2}$$

$$arg(\sqrt{y}+43-12\sqrt{7})<\pi$$

Solve for y (complex solution)

$$y=\left(-\sqrt{x}+43-12\sqrt{7}\right)^{2}$$

$$x=2857-1032\sqrt{7}\text{ or }arg(-\sqrt{x}+43-12\sqrt{7})\geq \pi$$