Solve vx+12=sqrt{x^2+2x-8}

$$v x + 12 = \sqrt { x ^ { 2 } + 2 x - 8 }$$

Solve for v (complex solution)

$$v=-\frac{-\sqrt{\left(x-2\right)\left(x+4\right)}+12}{x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Subtract 12 from both sides.

$$vx=\sqrt{x^{2}+2x-8}-12$$

The equation is in standard form.

$$xv=\sqrt{x^{2}+2x-8}-12$$

Divide both sides by x.

$$\frac{xv}{x}=\frac{\sqrt{\left(x-2\right)\left(x+4\right)}-12}{x}$$

Dividing by x undoes the multiplication by x.

$$v=\frac{\sqrt{\left(x-2\right)\left(x+4\right)}-12}{x}$$

Solve for v

$$v=-\frac{-\sqrt{\left(x-2\right)\left(x+4\right)}+12}{x}$$

$$x\leq -4\text{ or }x\geq 2$$

Steps for Solving Linear Equation

Subtract 12 from both sides.

$$vx=\sqrt{x^{2}+2x-8}-12$$

The equation is in standard form.

$$xv=\sqrt{x^{2}+2x-8}-12$$

Divide both sides by x.

$$\frac{xv}{x}=\frac{\sqrt{\left(x-2\right)\left(x+4\right)}-12}{x}$$

Dividing by x undoes the multiplication by x.

$$v=\frac{\sqrt{\left(x-2\right)\left(x+4\right)}-12}{x}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{153-24v-8v^{2}}+12v-1}{1-v^{2}}\text{; }x=\frac{-\sqrt{153-24v-8v^{2}}+12v-1}{1-v^{2}}\text{, }&v\neq 1\text{ and }v\neq -1\text{ and }arg(vx+12)<\pi \\x=\frac{76}{1-12v}\text{, }&\left(v=-1\text{ and }arg(12-x)<\pi \right)\text{ or }\left(v=1\text{ and }arg(x+12)<\pi \right)\\x=-\frac{12}{v}\text{, }&v=-6\text{ or }v=3\end{matrix}\right.$$