Factor with quadratic formula.
In general, given \(a{x}^{2}+bx+c\), the factored form is:
\[a(x-\frac{-b+\sqrt{{b}^{2}-4ac}}{2a})(x-\frac{-b-\sqrt{{b}^{2}-4ac}}{2a})\]
In this case, \(a=1\), \(b=-2\) and \(c=2\).
\[(x-\frac{2+\sqrt{{(-2)}^{2}-4\times 2}}{2})(x-\frac{2-\sqrt{{(-2)}^{2}-4\times 2}}{2})\]
Simplify.
\[(x-\frac{2+2\imath }{2})(x-\frac{2-2\imath }{2})\]
\[l\imath mx->1\times \sqrt[x-1]{(x-\frac{2+2\imath }{2})(x-\frac{2-2\imath }{2})}\]
Simplify \(\frac{2+2\imath }{2}\) to \(1+\frac{2\imath }{2}\).
\[l\imath mx->1\times \sqrt[x-1]{(x-(1+\frac{2\imath }{2}))(x-\frac{2-2\imath }{2})}\]
Cancel \(2\).
\[l\imath mx->1\times \sqrt[x-1]{(x-(1+\imath ))(x-\frac{2-2\imath }{2})}\]
Remove parentheses.
\[l\imath mx->1\times \sqrt[x-1]{(x-1-\imath )(x-\frac{2-2\imath }{2})}\]
Simplify \(\frac{2-2\imath }{2}\) to \(1-\frac{2\imath }{2}\).
\[l\imath mx->1\times \sqrt[x-1]{(x-1-\imath )(x-(1-\frac{2\imath }{2}))}\]
Cancel \(2\).
\[l\imath mx->1\times \sqrt[x-1]{(x-1-\imath )(x-(1-\imath ))}\]
Remove parentheses.
\[l\imath mx->1\times \sqrt[x-1]{(x-1-\imath )(x-1+\imath )}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[l\imath mx->1\times \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\]
Simplify \(1\times \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\) to \(\sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\).
\[l\imath mx->\sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\]
Regroup terms.
\[-+l\imath mx>\sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\]
Divide both sides by \(\imath \).
\[-+lmx>\frac{\sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{\imath }\]
Rationalize the denominator: \(\frac{\sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{\imath } \cdot \frac{\imath }{\imath }=-\sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\imath \).
\[-+lmx>-\sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\imath \]
Regroup terms.
\[-+lmx>-\imath \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }\]
Divide both sides by \(m\).
\[-+lx>-\frac{\imath \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{m}\]
Divide both sides by \(x\).
\[-+l>-\frac{\frac{\imath \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{m}}{x}\]
Simplify \(\frac{\frac{\imath \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{m}}{x}\) to \(\frac{\imath \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{mx}\).
\[-+l>-\frac{\imath \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{mx}\]
Multiply both sides by \(-1\).
\[l<\frac{\imath \sqrt[x-1]{x-1-\imath }\sqrt[x-1]{x-1+\imath }}{mx}\]
l<(IM*(x-1-IM)^(1/(x-1))*(x-1+IM)^(1/(x-1)))/(m*x)
