Solve ksqrt{x-2}-x-2k-10=0

$$k \sqrt { x - 2 } - x - 2 k - 10 = 0$$

Solve for k

$$k=\frac{\left(\sqrt{x-2}+2\right)\left(x+10\right)}{x-6}$$

$$x\neq 6\text{ and }x\geq 2$$

Steps for Solving Linear Equation

Add x to both sides. Anything plus zero gives itself.

$$k\sqrt{x-2}-2k-10=x$$

Add 10 to both sides.

$$k\sqrt{x-2}-2k=x+10$$

Combine all terms containing k.

$$\left(\sqrt{x-2}-2\right)k=x+10$$

Divide both sides by \sqrt{x-2}-2.

$$\frac{\left(\sqrt{x-2}-2\right)k}{\sqrt{x-2}-2}=\frac{x+10}{\sqrt{x-2}-2}$$

Dividing by \sqrt{x-2}-2 undoes the multiplication by \sqrt{x-2}-2.

$$k=\frac{x+10}{\sqrt{x-2}-2}$$

Divide x+10 by \sqrt{x-2}-2.

$$k=\frac{\left(\sqrt{x-2}+2\right)\left(x+10\right)}{x-6}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{k\sqrt{\left(k-12\right)\left(k+4\right)}+k^{2}-4k-20}{2}\text{, }&\frac{\sqrt{1-\frac{8k+48}{k^{2}}}k^{2}}{2}+\frac{k^{2}}{2}-2k\geq 12\text{ and }k\geq 12\\x=\frac{-\sqrt{\left(k-12\right)\left(k+4\right)}k^{2}+|k|k^{2}-4k|k|-20|k|}{2|k|}\text{, }&-\frac{\sqrt{1-\frac{8k+48}{k^{2}}}k^{2}}{2}+\frac{k^{2}}{2}-2k\geq 12\text{ and }\left(k\geq 12\text{ or }k\leq -6\right)\end{matrix}\right.$$