Solve for S_0 (complex solution)
\left\{\begin{matrix}S_{0}=-\frac{x^{2}-4}{fx}\text{, }&x\neq 0\text{ and }f\neq 0\\S_{0}\in \mathrm{C}\text{, }&\left(x=-2\text{ or }x=2\right)\text{ and }f=0\end{matrix}\right.
Solve for f (complex solution)
\left\{\begin{matrix}f=-\frac{x^{2}-4}{S_{0}x}\text{, }&x\neq 0\text{ and }S_{0}\neq 0\\f\in \mathrm{C}\text{, }&\left(x=-2\text{ or }x=2\right)\text{ and }S_{0}=0\end{matrix}\right.
Solve for S_0
\left\{\begin{matrix}S_{0}=-\frac{x^{2}-4}{fx}\text{, }&x\neq 0\text{ and }f\neq 0\\S_{0}\in \mathrm{R}\text{, }&f=0\text{ and }|x|=2\end{matrix}\right.
Solve for f
\left\{\begin{matrix}f=-\frac{x^{2}-4}{S_{0}x}\text{, }&x\neq 0\text{ and }S_{0}\neq 0\\f\in \mathrm{R}\text{, }&S_{0}=0\text{ and }|x|=2\end{matrix}\right.
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fxS_{0}=4-x^{2}
The equation is in standard form.
\frac{fxS_{0}}{fx}=\frac{4-x^{2}}{fx}
Divide both sides by fx.
S_{0}=\frac{4-x^{2}}{fx}
Dividing by fx undoes the multiplication by fx.
S_{0}=\frac{\left(2-x\right)\left(x+2\right)}{fx}
Divide 4-x^{2} by fx.
S_{0}xf=4-x^{2}
The equation is in standard form.
\frac{S_{0}xf}{S_{0}x}=\frac{4-x^{2}}{S_{0}x}
Divide both sides by S_{0}x.
f=\frac{4-x^{2}}{S_{0}x}
Dividing by S_{0}x undoes the multiplication by S_{0}x.
f=\frac{\left(2-x\right)\left(x+2\right)}{S_{0}x}
Divide 4-x^{2} by S_{0}x.
fxS_{0}=4-x^{2}
The equation is in standard form.
\frac{fxS_{0}}{fx}=\frac{4-x^{2}}{fx}
Divide both sides by fx.
S_{0}=\frac{4-x^{2}}{fx}
Dividing by fx undoes the multiplication by fx.
S_{0}=\frac{\left(2-x\right)\left(x+2\right)}{fx}
Divide 4-x^{2} by fx.
S_{0}xf=4-x^{2}
The equation is in standard form.
\frac{S_{0}xf}{S_{0}x}=\frac{4-x^{2}}{S_{0}x}
Divide both sides by S_{0}x.
f=\frac{4-x^{2}}{S_{0}x}
Dividing by S_{0}x undoes the multiplication by S_{0}x.
f=\frac{\left(2-x\right)\left(x+2\right)}{S_{0}x}
Divide 4-x^{2} by S_{0}x.
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