Solve for \(z\) in \(+5Re(\frac{5+6\imath }{\imath })+{(\imath -z)}^{2}=-\frac{6-6\imath }{1+\imath }+{\imath }^{26}+{\imath }^{10}\).
Solve for \(z\).
\[+5Re(\frac{5+6\imath }{\imath })+{(\imath -z)}^{2}=-\frac{6-6\imath }{1+\imath }+{\imath }^{26}+{\imath }^{10}\]
Remove parentheses.
\[5Re(\frac{5+6\imath }{\imath })+{(\imath -z)}^{2}=-\frac{6-6\imath }{1+\imath }+{\imath }^{26}+{\imath }^{10}\]
Rationalize the denominator: \(\frac{5+6\imath }{\imath } \cdot \frac{\imath }{\imath }=-(5+6\imath )\imath \).
\[5Re(-(5+6\imath )\imath )+{(\imath -z)}^{2}=-\frac{6-6\imath }{1+\imath }+{\imath }^{26}+{\imath }^{10}\]
Regroup terms.
\[5Re(-(5+6\imath )\imath )+{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }+{\imath }^{26}+{\imath }^{10}\]
Use Power Reduction Rule: \({i}^{n}={i}^{n\text{ mod }4}\).
\[5Re(-(5+6\imath )\imath )+{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }+{\imath }^{2}+{\imath }^{10}\]
Use Square Rule: \({i}^{2}=-1\).
\[5Re(-(5+6\imath )\imath )+{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }-1+{\imath }^{10}\]
Use Power Reduction Rule: \({i}^{n}={i}^{n\text{ mod }4}\).
\[5Re(-(5+6\imath )\imath )+{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }-1+{\imath }^{2}\]
Use Square Rule: \({i}^{2}=-1\).
\[5Re(-(5+6\imath )\imath )+{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }-1-1\]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[5Re(-(5+6\imath )\imath )+{(-z+\imath )}^{2}=-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1\]
Cancel \(1-\imath \).
\[5Re(-(5+6\imath )\imath )+{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }-1-1\]
Subtract \(5Re(-(5+6\imath )\imath )\) from both sides.
\[{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )\]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[{(-z+\imath )}^{2}=-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )\]
Cancel \(1-\imath \).
\[{(-z+\imath )}^{2}=-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )\]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[{(-z+\imath )}^{2}=-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )\]
Take the square root of both sides.
\[-z+\imath =\pm \sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}\]
Cancel \(1-\imath \).
\[-z+\imath =\pm \sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[-z+\imath =\pm \sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}\]
Cancel \(1-\imath \).
\[-z+\imath =\pm \sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\]
Break down the problem into these 2 equations.
\[-z+\imath =\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\]
\[-z+\imath =-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\]
Solve the 1st equation: \(-z+\imath =\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\).
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[-z+\imath =\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}\]
Cancel \(1-\imath \).
\[-z+\imath =\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[-z+\imath =\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}\]
Subtract \(\imath \) from both sides.
\[-z=\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Cancel \(1-\imath \).
\[-z=\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[-z=\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Cancel \(1-\imath \).
\[-z=\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Multiply both sides by \(-1\).
\[z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[z=-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Cancel \(1-\imath \).
\[z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[z=-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
\[z=-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Solve the 2nd equation: \(-z+\imath =-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\).
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[-z+\imath =-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}\]
Cancel \(1-\imath \).
\[-z+\imath =-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}\]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[-z+\imath =-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}\]
Subtract \(\imath \) from both sides.
\[-z=-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Cancel \(1-\imath \).
\[-z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[-z=-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Cancel \(1-\imath \).
\[-z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}-\imath \]
Multiply both sides by \(-1\).
\[z=\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[z=\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Cancel \(1-\imath \).
\[z=\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[z=\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
\[z=\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Collect all solutions.
\[z=-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath ,\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Cancel \(1-\imath \).
\[z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath ,\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Rationalize the denominator: \(\frac{6-6\imath }{1+\imath } \cdot \frac{1-\imath }{1-\imath }=\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}\).
\[z=-\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath ,\sqrt{-\frac{(6-6\imath )(1-\imath )}{(1+\imath )(1-\imath )}-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
Cancel \(1-\imath \).
\[z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath ,\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
\[z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath ,\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \]
\[\begin{aligned}&z=-\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath ,\sqrt{-\frac{6-6\imath }{1+\imath }-1-1-5Re(-(5+6\imath )\imath )}+\imath \\&z=1+9\imath \end{aligned}\]
z=-sqrt(-(6-6*IM)/(1+IM)-1-1-5*Re(-(5+6*IM)*IM))+IM,sqrt(-(6-6*IM)/(1+IM)-1-1-5*Re(-(5+6*IM)*IM))+IM;z=1+9*IM
