\sqrt((4x^(2)+25))

on Jun 03, 2023 at 4:33 am

Evaluate the expression: $$\sqrt{4 \cdot {x}^{2} + 25}$$

#### Evaluated

$$\displaystyle \sqrt{4 \cdot {x}^{2} + 25} = \sqrt{4 x^{2} + 25}$$

#### Roots

$\sqrt{4 \cdot {x}^{2} + 25} = i \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25\right)^{2} + 64 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)} + \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25\right)^{2} + 64 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)} \approx 5.0 i \left(\left(0.16 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 0.16 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right)^{2} + 0.1024 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)} + 5.0 \left(\left(0.16 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 0.16 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right)^{2} + 0.1024 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)}$$\sqrt{4 \cdot {x}^{2} + 25} = i \left(- \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25\right)^{2} + 64 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)}\right) - \sqrt[4]{\left(4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25\right)^{2} + 64 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)} \approx - 5.0 i \left(\left(0.16 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 0.16 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right)^{2} + 0.1024 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{0.25} \sin{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)} - 5.0 \left(\left(0.16 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 0.16 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 1\right)^{2} + 0.1024 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{0.25} \cos{\left(\frac{\operatorname{atan_{2}}{\left(8 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)},4 \left(\operatorname{re}{\left(x\right)}\right)^{2} - 4 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 25 \right)}}{2} \right)}$