Regroup terms.
\[3fP\imath x+\frac{1}{x}=5f\imath ndf\imath nd{x}^{4}+\frac{1}{{x}^{4}}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[3fP\imath x+\frac{1}{x}=5{f}^{2}{\imath }^{2}{n}^{2}{d}^{2}{x}^{4}+\frac{1}{{x}^{4}}\]
Use Square Rule: \({i}^{2}=-1\).
\[3fP\imath x+\frac{1}{x}=5{f}^{2}\times -1\times {n}^{2}{d}^{2}{x}^{4}+\frac{1}{{x}^{4}}\]
Simplify \(5{f}^{2}\times -1\times {n}^{2}{d}^{2}{x}^{4}\) to \(-5{f}^{2}{n}^{2}{d}^{2}{x}^{4}\).
\[3fP\imath x+\frac{1}{x}=-5{f}^{2}{n}^{2}{d}^{2}{x}^{4}+\frac{1}{{x}^{4}}\]
Regroup terms.
\[\frac{1}{x}+3fP\imath x=-5{f}^{2}{n}^{2}{d}^{2}{x}^{4}+\frac{1}{{x}^{4}}\]
Subtract \(\frac{1}{{x}^{4}}\) from both sides.
\[\frac{1}{x}+3fP\imath x-\frac{1}{{x}^{4}}=-5{f}^{2}{n}^{2}{d}^{2}{x}^{4}\]
Divide both sides by \(-5\).
\[-\frac{\frac{1}{x}+3fP\imath x-\frac{1}{{x}^{4}}}{5}={f}^{2}{n}^{2}{d}^{2}{x}^{4}\]
Simplify \(\frac{\frac{1}{x}+3fP\imath x-\frac{1}{{x}^{4}}}{5}\) to \(\frac{\frac{1}{x}}{5}+\frac{3fP\imath x}{5}-\frac{\frac{1}{{x}^{4}}}{5}\).
\[-(\frac{\frac{1}{x}}{5}+\frac{3fP\imath x}{5}-\frac{\frac{1}{{x}^{4}}}{5})={f}^{2}{n}^{2}{d}^{2}{x}^{4}\]
Simplify \(\frac{\frac{1}{x}}{5}\) to \(\frac{1}{5x}\).
\[-(\frac{1}{5x}+\frac{3fP\imath x}{5}-\frac{\frac{1}{{x}^{4}}}{5})={f}^{2}{n}^{2}{d}^{2}{x}^{4}\]
Simplify \(\frac{\frac{1}{{x}^{4}}}{5}\) to \(\frac{1}{5{x}^{4}}\).
\[-(\frac{1}{5x}+\frac{3fP\imath x}{5}-\frac{1}{5{x}^{4}})={f}^{2}{n}^{2}{d}^{2}{x}^{4}\]
Remove parentheses.
\[-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}={f}^{2}{n}^{2}{d}^{2}{x}^{4}\]
Divide both sides by \({f}^{2}\).
\[\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}}={n}^{2}{d}^{2}{x}^{4}\]
Divide both sides by \({d}^{2}\).
\[\frac{\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}}}{{d}^{2}}={n}^{2}{x}^{4}\]
Simplify \(\frac{\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}}}{{d}^{2}}\) to \(\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}}\).
\[\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}}={n}^{2}{x}^{4}\]
Divide both sides by \({x}^{4}\).
\[\frac{\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}}}{{x}^{4}}={n}^{2}\]
Simplify \(\frac{\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}}}{{x}^{4}}\) to \(\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}{x}^{4}}\).
\[\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}{x}^{4}}={n}^{2}\]
Take the square root of both sides.
\[\pm \sqrt{\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}{x}^{4}}}=n\]
Switch sides.
\[n=\pm \sqrt{\frac{-\frac{1}{5x}-\frac{3fP\imath x}{5}+\frac{1}{5{x}^{4}}}{{f}^{2}{d}^{2}{x}^{4}}}\]
n=sqrt((-1/(5*x)-(3*fP*IM*x)/5+1/(5*x^4))/(f^2*d^2*x^4)),-sqrt((-1/(5*x)-(3*fP*IM*x)/5+1/(5*x^4))/(f^2*d^2*x^4))
