Expand.
\[={r}^{2}-{(overl\imath neVb)}^{2}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[={r}^{2}-{o}^{2}{v}^{2}{e}^{2}{r}^{2}{l}^{2}{\imath }^{2}{n}^{2}{eVb}^{2}\]
Use Square Rule: \({i}^{2}=-1\).
\[={r}^{2}-{o}^{2}{v}^{2}{e}^{2}{r}^{2}{l}^{2}\times -1\times {n}^{2}{eVb}^{2}\]
Simplify \({o}^{2}{v}^{2}{e}^{2}{r}^{2}{l}^{2}\times -1\times {n}^{2}{eVb}^{2}\) to \({o}^{2}{v}^{2}{e}^{2}{r}^{2}{l}^{2}\times -{n}^{2}{eVb}^{2}\).
\[={r}^{2}-{o}^{2}{v}^{2}{e}^{2}{r}^{2}{l}^{2}\times -{n}^{2}{eVb}^{2}\]
Regroup terms.
\[={r}^{2}-(-{e}^{2}{eVb}^{2}{o}^{2}{v}^{2}{r}^{2}{l}^{2}{n}^{2})\]
Subtract \({r}^{2}\) from both sides.
\[-{r}^{2}={e}^{2}{eVb}^{2}{o}^{2}{v}^{2}{r}^{2}{l}^{2}{n}^{2}\]
Divide both sides by \({e}^{2}\).
\[-\frac{{r}^{2}}{{e}^{2}}={eVb}^{2}{o}^{2}{v}^{2}{r}^{2}{l}^{2}{n}^{2}\]
Divide both sides by \({eVb}^{2}\).
\[-\frac{\frac{{r}^{2}}{{e}^{2}}}{{eVb}^{2}}={o}^{2}{v}^{2}{r}^{2}{l}^{2}{n}^{2}\]
Simplify \(\frac{\frac{{r}^{2}}{{e}^{2}}}{{eVb}^{2}}\) to \(\frac{{r}^{2}}{{e}^{2}{eVb}^{2}}\).
\[-\frac{{r}^{2}}{{e}^{2}{eVb}^{2}}={o}^{2}{v}^{2}{r}^{2}{l}^{2}{n}^{2}\]
Divide both sides by \({o}^{2}\).
\[-\frac{\frac{{r}^{2}}{{e}^{2}{eVb}^{2}}}{{o}^{2}}={v}^{2}{r}^{2}{l}^{2}{n}^{2}\]
Simplify \(\frac{\frac{{r}^{2}}{{e}^{2}{eVb}^{2}}}{{o}^{2}}\) to \(\frac{{r}^{2}}{{e}^{2}{eVb}^{2}{o}^{2}}\).
\[-\frac{{r}^{2}}{{e}^{2}{eVb}^{2}{o}^{2}}={v}^{2}{r}^{2}{l}^{2}{n}^{2}\]
Divide both sides by \({r}^{2}\).
\[-\frac{\frac{{r}^{2}}{{e}^{2}{eVb}^{2}{o}^{2}}}{{r}^{2}}={v}^{2}{l}^{2}{n}^{2}\]
Simplify \(\frac{\frac{{r}^{2}}{{e}^{2}{eVb}^{2}{o}^{2}}}{{r}^{2}}\) to \(\frac{{r}^{2}}{{e}^{2}{eVb}^{2}{o}^{2}{r}^{2}}\).
\[-\frac{{r}^{2}}{{e}^{2}{eVb}^{2}{o}^{2}{r}^{2}}={v}^{2}{l}^{2}{n}^{2}\]
Cancel \({r}^{2}\).
\[-\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}}={v}^{2}{l}^{2}{n}^{2}\]
Divide both sides by \({l}^{2}\).
\[-\frac{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}}}{{l}^{2}}={v}^{2}{n}^{2}\]
Simplify \(\frac{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}}}{{l}^{2}}\) to \(\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}}\).
\[-\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}}={v}^{2}{n}^{2}\]
Divide both sides by \({n}^{2}\).
\[-\frac{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}}}{{n}^{2}}={v}^{2}\]
Simplify \(\frac{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}}}{{n}^{2}}\) to \(\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}\).
\[-\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}={v}^{2}\]
Take the square root of both sides.
\[\pm \sqrt{-\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}}=v\]
Simplify \(\sqrt{-\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}}\) to \(\sqrt{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}}\imath \).
\[\pm \sqrt{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}}\imath =v\]
Regroup terms.
\[\pm \imath \sqrt{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}}=v\]
Switch sides.
\[v=\pm \imath \sqrt{\frac{1}{{e}^{2}{eVb}^{2}{o}^{2}{l}^{2}{n}^{2}}}\]
v=IM*sqrt(1/(e^2*eVb^2*o^2*l^2*n^2)),-IM*sqrt(1/(e^2*eVb^2*o^2*l^2*n^2))
