Solve U=9PR^2h-9Pr^2h

$$U = 9 P R ^ { 2 } h - 9 P r ^ { 2 } h$$

Solve for P (complex solution)

$$\left\{\begin{matrix}P=-\frac{U}{9h\left(r^{2}-R^{2}\right)}\text{, }&R\neq -r\text{ and }R\neq r\text{ and }h\neq 0\\P\in \mathrm{C}\text{, }&\left(R=-r\text{ or }R=r\text{ or }h=0\right)\text{ and }U=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$9PR^{2}h-9Pr^{2}h=U$$

Combine all terms containing P.

$$\left(9R^{2}h-9r^{2}h\right)P=U$$

The equation is in standard form.

$$\left(9hR^{2}-9hr^{2}\right)P=U$$

Divide both sides by 9R^{2}h-9r^{2}h.

$$\frac{\left(9hR^{2}-9hr^{2}\right)P}{9hR^{2}-9hr^{2}}=\frac{U}{9hR^{2}-9hr^{2}}$$

Dividing by 9R^{2}h-9r^{2}h undoes the multiplication by 9R^{2}h-9r^{2}h.

$$P=\frac{U}{9hR^{2}-9hr^{2}}$$

Divide U by 9R^{2}h-9r^{2}h.

$$P=\frac{U}{9h\left(R^{2}-r^{2}\right)}$$

Solve for P

$$\left\{\begin{matrix}P=-\frac{U}{9h\left(r^{2}-R^{2}\right)}\text{, }&h\neq 0\text{ and }|R|\neq |r|\\P\in \mathrm{R}\text{, }&\left(|R|=|r|\text{ or }h=0\right)\text{ and }U=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$9PR^{2}h-9Pr^{2}h=U$$

Combine all terms containing P.

$$\left(9R^{2}h-9r^{2}h\right)P=U$$

The equation is in standard form.

$$\left(9hR^{2}-9hr^{2}\right)P=U$$

Divide both sides by 9R^{2}h-9r^{2}h.

$$\frac{\left(9hR^{2}-9hr^{2}\right)P}{9hR^{2}-9hr^{2}}=\frac{U}{9hR^{2}-9hr^{2}}$$

Dividing by 9R^{2}h-9r^{2}h undoes the multiplication by 9R^{2}h-9r^{2}h.

$$P=\frac{U}{9hR^{2}-9hr^{2}}$$

Divide U by 9R^{2}h-9r^{2}h.

$$P=\frac{U}{9h\left(R^{2}-r^{2}\right)}$$

Solve for R (complex solution)

$$\left\{\begin{matrix}R=-\frac{\sqrt{\frac{9Phr^{2}+U}{Ph}}}{3}\text{; }R=\frac{\sqrt{\frac{9Phr^{2}+U}{Ph}}}{3}\text{, }&h\neq 0\text{ and }P\neq 0\\R\in \mathrm{C}\text{, }&\left(P=0\text{ or }h=0\right)\text{ and }U=0\end{matrix}\right.$$