Math Problem Solver

Roots

\[\sqrt{9 + 4 \cdot {a}^{2}} = i \sqrt[4]{\left(4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9\right)^{2} + 64 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)} + \sqrt[4]{\left(4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9\right)^{2} + 64 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)} \approx 3.0 i \left(\left(0.444444444444444 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.444444444444444 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 1\right)^{2} + 0.790123456790123 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)} + 3.0 \left(\left(0.444444444444444 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.444444444444444 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 1\right)^{2} + 0.790123456790123 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)}\]\[\sqrt{9 + 4 \cdot {a}^{2}} = i \left(- \sqrt[4]{\left(4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9\right)^{2} + 64 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)}\right) - \sqrt[4]{\left(4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9\right)^{2} + 64 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)} \approx - 3.0 i \left(\left(0.444444444444444 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.444444444444444 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 1\right)^{2} + 0.790123456790123 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)} - 3.0 \left(\left(0.444444444444444 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 0.444444444444444 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 1\right)^{2} + 0.790123456790123 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},4 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 \right)}}{2} \right)}\]