Math Problem Solver

Roots

\[\sqrt{2 \cdot {a}^{2} - {r}^{2}} = i \sqrt[4]{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)} + \sqrt[4]{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)} \approx 2.0 i \left(\left(\operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.5 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + 0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.5 \left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.5 \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)} + 2.0 \left(\left(\operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.5 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + 0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.5 \left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.5 \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)}\]\[\sqrt{2 \cdot {a}^{2} - {r}^{2}} = i \left(- \sqrt[4]{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)}\right) - \sqrt[4]{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)} \approx - 2.0 i \left(\left(\operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.5 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + 0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.5 \left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.5 \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)} - 2.0 \left(\left(\operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 0.5 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)}\right)^{2} + 0.25 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.5 \left(\operatorname{re}{\left(r\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 0.5 \left(\operatorname{im}{\left(r\right)}\right)^{2}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 2 \operatorname{re}{\left(r\right)} \operatorname{im}{\left(r\right)},2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{re}{\left(r\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(r\right)}\right)^{2} \right)}}{2} \right)}\]