Solve R=sqrt{ax-p}(x)

$$R = \sqrt { a x - p } ( x )$$

Solve for a

$$\left\{\begin{matrix}a=\frac{px^{2}+R^{2}}{x^{3}}\text{, }&\left(R\geq 0\text{ and }x>0\right)\text{ or }\left(R\leq 0\text{ and }x<0\right)\\a\in \mathrm{R}\text{, }&R=0\text{ and }p\leq 0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Radical Equations

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

$$\sqrt{ax-p}x=R$$

Divide both sides by x.

$$\frac{x\sqrt{xa-p}}{x}=\frac{R}{x}$$

Dividing by x undoes the multiplication by x.

$$\sqrt{xa-p}=\frac{R}{x}$$

Square both sides of the equation.

$$xa-p=\frac{R^{2}}{x^{2}}$$

Subtract -p from both sides of the equation.

$$xa-p-\left(-p\right)=\frac{R^{2}}{x^{2}}-\left(-p\right)$$

Subtracting -p from itself leaves 0.

$$xa=\frac{R^{2}}{x^{2}}-\left(-p\right)$$

Subtract -p from \frac{R^{2}}{x^{2}}.

$$xa=p+\frac{R^{2}}{x^{2}}$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{p+\frac{R^{2}}{x^{2}}}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{p+\frac{R^{2}}{x^{2}}}{x}$$

Divide p+\frac{R^{2}}{x^{2}} by x.

$$a=\frac{p}{x}+\frac{R^{2}}{x^{3}}$$

Solve for R (complex solution)

$$R=x\sqrt{ax-p}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{px^{2}+R^{2}}{x^{3}}\text{, }&x\neq 0\text{ and }\left(|arg(\sqrt{\frac{R^{2}}{x^{2}}}x)-arg(R)|<\pi \text{ or }R=0\right)\\a\in \mathrm{C}\text{, }&R=0\text{ and }x=0\end{matrix}\right.$$

Solve for R

$$R=x\sqrt{ax-p}$$

$$p\leq ax$$