Solve for A
\left\{\begin{matrix}\\A=0\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&|B|=\sqrt{2}|C|\end{matrix}\right.
Solve for B
\left\{\begin{matrix}\\B=\sqrt{2}C\text{; }B=-\sqrt{2}C\text{, }&\text{unconditionally}\\B\in \mathrm{R}\text{, }&A=0\end{matrix}\right.
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AB^{2}-2AC^{2}=0
Subtract 2AC^{2} from both sides.
\left(B^{2}-2C^{2}\right)A=0
Combine all terms containing A.
A=0
Divide 0 by B^{2}-2C^{2}.
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