find square root of (x/y+y/x)^2-4(x/y-y/x)
You asked:
MathBot Answer:
Evaluated
\(\displaystyle \sqrt{{\left( \frac{x}{y} + \frac{y}{x} \right)}^{2} - 4 \cdot \left(\frac{x}{y} - \frac{y}{x}\right)} = \sqrt{- \frac{4 x}{y} + \left(\frac{x}{y} + \frac{y}{x}\right)^{2} + \frac{4 y}{x}} \)
Expanded
\[\sqrt{{\left( \frac{x}{y} + \frac{y}{x} \right)}^{2} - 4 \cdot \left(\frac{x}{y} - \frac{y}{x}\right)} = \sqrt{\frac{x^{2}}{y^{2}} - \frac{4 x}{y} + 2 + \frac{4 y}{x} + \frac{y^{2}}{x^{2}}}\]
Factored
\[\sqrt{{\left( \frac{x}{y} + \frac{y}{x} \right)}^{2} - 4 \cdot \left(\frac{x}{y} - \frac{y}{x}\right)} = \sqrt{\frac{x^{2}}{y^{2}} - \frac{4 x}{y} + 2 + \frac{4 y}{x} + \frac{y^{2}}{x^{2}}}\]
Roots
\[\sqrt{{\left( \frac{x}{y} + \frac{y}{x} \right)}^{2} - 4 \cdot \left(\frac{x}{y} - \frac{y}{x}\right)} = i \sqrt[4]{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)} \approx 2.0 i \left(\left(0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + \operatorname{im}{\left(\frac{y}{x}\right)} - \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(0.25 \left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + \operatorname{re}{\left(\frac{y}{x}\right)} + 0.25 \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - \operatorname{re}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 0.5 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)} + 2.0 \left(\left(0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + \operatorname{im}{\left(\frac{y}{x}\right)} - \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(0.25 \left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + \operatorname{re}{\left(\frac{y}{x}\right)} + 0.25 \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - \operatorname{re}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 0.5 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)}\]\[\sqrt{{\left( \frac{x}{y} + \frac{y}{x} \right)}^{2} - 4 \cdot \left(\frac{x}{y} - \frac{y}{x}\right)} = i \left(- \sqrt[4]{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)}\right) - \sqrt[4]{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)} \approx - 2.0 i \left(\left(0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + \operatorname{im}{\left(\frac{y}{x}\right)} - \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(0.25 \left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + \operatorname{re}{\left(\frac{y}{x}\right)} + 0.25 \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - \operatorname{re}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 0.5 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)} - 2.0 \left(\left(0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 0.5 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + \operatorname{im}{\left(\frac{y}{x}\right)} - \operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} + \left(0.25 \left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 0.5 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + \operatorname{re}{\left(\frac{y}{x}\right)} + 0.25 \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - \operatorname{re}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 0.5 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - 0.25 \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{y}{x}\right)} + 2 \operatorname{re}{\left(\frac{x}{y}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} + 4 \operatorname{im}{\left(\frac{y}{x}\right)} - 4 \operatorname{im}{\left(\frac{x}{y}\right)},\left(\operatorname{re}{\left(\frac{y}{x}\right)}\right)^{2} + 2 \operatorname{re}{\left(\frac{y}{x}\right)} \operatorname{re}{\left(\frac{x}{y}\right)} + 4 \operatorname{re}{\left(\frac{y}{x}\right)} + \left(\operatorname{re}{\left(\frac{x}{y}\right)}\right)^{2} - 4 \operatorname{re}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{y}{x}\right)}\right)^{2} - 2 \operatorname{im}{\left(\frac{y}{x}\right)} \operatorname{im}{\left(\frac{x}{y}\right)} - \left(\operatorname{im}{\left(\frac{x}{y}\right)}\right)^{2} \right)}}{2} \right)}\]