Solve AB^{4}-AC^4=BC^2(AB^2-AC^2)

$$A B ^ { 4 } - A C ^ { 4 } = B C ^ { 2 } ( A B ^ { 2 } - A C ^ { 2 } )$$

Solve for A (complex solution)

$$\left\{\begin{matrix}\\A=0\text{, }&\text{unconditionally}\\A\in \mathrm{C}\text{, }&B=\frac{\sqrt{C^{4}-4C^{2}}+C^{2}}{2}\text{ or }B=\frac{-\sqrt{C^{4}-4C^{2}}+C^{2}}{2}\text{ or }B=-C\text{ or }B=C\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract BC^{2}\left(AB^{2}-AC^{2}\right) from both sides.

$$AB^{4}-AC^{4}-BC^{2}\left(AB^{2}-AC^{2}\right)=0$$

Use the distributive property to multiply -BC^{2} by AB^{2}-AC^{2}.

$$AB^{4}-AC^{4}-C^{2}AB^{3}+BC^{4}A=0$$

Combine all terms containing A.

$$\left(B^{4}-C^{4}-C^{2}B^{3}+BC^{4}\right)A=0$$

The equation is in standard form.

$$\left(B^{4}+BC^{4}-C^{4}-C^{2}B^{3}\right)A=0$$

Divide 0 by B^{4}-C^{4}-C^{2}B^{3}+BC^{4}.

$$A=0$$

Solve for A

$$\left\{\begin{matrix}\\A=0\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&|B|=|C|\text{ or }\left(B=\frac{C\left(\sqrt{C^{2}-4}+C\right)}{2}\text{ and }|C|\geq 2\right)\text{ or }\left(B=\frac{C\left(-\sqrt{C^{2}-4}+C\right)}{2}\text{ and }|C|\geq 2\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract BC^{2}\left(AB^{2}-AC^{2}\right) from both sides.

$$AB^{4}-AC^{4}-BC^{2}\left(AB^{2}-AC^{2}\right)=0$$

Use the distributive property to multiply -BC^{2} by AB^{2}-AC^{2}.

$$AB^{4}-AC^{4}-C^{2}AB^{3}+BC^{4}A=0$$

Combine all terms containing A.

$$\left(B^{4}-C^{4}-C^{2}B^{3}+BC^{4}\right)A=0$$

The equation is in standard form.

$$\left(B^{4}+BC^{4}-C^{4}-C^{2}B^{3}\right)A=0$$

Divide 0 by B^{4}-C^{4}-C^{2}B^{3}+BC^{4}.

$$A=0$$

Solve for B (complex solution)

$$\left\{\begin{matrix}\\B=C\text{; }B=-C\text{; }B=\frac{-\sqrt{C^{2}\left(C^{2}-4\right)}+C^{2}}{2}\text{; }B=\frac{\sqrt{C^{2}\left(C^{2}-4\right)}+C^{2}}{2}\text{, }&\text{unconditionally}\\B\in \mathrm{C}\text{, }&A=0\end{matrix}\right.$$

Solve for B

$$\left\{\begin{matrix}\\B=C\text{; }B=-C\text{, }&\text{unconditionally}\\B=\frac{C\left(\sqrt{C^{2}-4}+C\right)}{2}\text{; }B=\frac{C\left(-\sqrt{C^{2}-4}+C\right)}{2}\text{, }&|C|\geq 2\\B\in \mathrm{R}\text{, }&A=0\end{matrix}\right.$$