Solve AR^2+PB^2=PR^2+AB^2

$$A R ^ { 2 } + P B ^ { 2 } = P R ^ { 2 } + A B ^ { 2 }$$

Solve for A (complex solution)

$$\left\{\begin{matrix}\\A=P\text{, }&\text{unconditionally}\\A\in \mathrm{C}\text{, }&R=-B\text{ or }R=B\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract AB^{2} from both sides.

$$AR^{2}+PB^{2}-AB^{2}=PR^{2}$$

Subtract PB^{2} from both sides.

$$AR^{2}-AB^{2}=PR^{2}-PB^{2}$$

Combine all terms containing A.

$$\left(R^{2}-B^{2}\right)A=PR^{2}-PB^{2}$$

Divide both sides by R^{2}-B^{2}.

$$\frac{\left(R^{2}-B^{2}\right)A}{R^{2}-B^{2}}=\frac{P\left(R-B\right)\left(B+R\right)}{R^{2}-B^{2}}$$

Dividing by R^{2}-B^{2} undoes the multiplication by R^{2}-B^{2}.

$$A=\frac{P\left(R-B\right)\left(B+R\right)}{R^{2}-B^{2}}$$

Divide P\left(R-B\right)\left(R+B\right) by R^{2}-B^{2}.

$$A=P$$

Solve for A

$$\left\{\begin{matrix}\\A=P\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&|R|=|B|\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract AB^{2} from both sides.

$$AR^{2}+PB^{2}-AB^{2}=PR^{2}$$

Subtract PB^{2} from both sides.

$$AR^{2}-AB^{2}=PR^{2}-PB^{2}$$

Combine all terms containing A.

$$\left(R^{2}-B^{2}\right)A=PR^{2}-PB^{2}$$

Divide both sides by R^{2}-B^{2}.

$$\frac{\left(R^{2}-B^{2}\right)A}{R^{2}-B^{2}}=\frac{P\left(R-B\right)\left(B+R\right)}{R^{2}-B^{2}}$$

Dividing by R^{2}-B^{2} undoes the multiplication by R^{2}-B^{2}.

$$A=\frac{P\left(R-B\right)\left(B+R\right)}{R^{2}-B^{2}}$$

Divide P\left(R-B\right)\left(R+B\right) by R^{2}-B^{2}.

$$A=P$$

Solve for B (complex solution)

$$\left\{\begin{matrix}\\B=-R\text{; }B=R\text{, }&\text{unconditionally}\\B\in \mathrm{C}\text{, }&A=P\end{matrix}\right.$$

Solve for B

$$\left\{\begin{matrix}\\B=-R\text{; }B=R\text{, }&\text{unconditionally}\\B\in \mathrm{R}\text{, }&A=P\end{matrix}\right.$$