Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(y^{4} - y^{2}\right) + 5$$
Para eso usemos la fórmula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = -1$$
$$c = 5$$
Entonces
$$m = - \frac{1}{2}$$
$$n = \frac{19}{4}$$
Pues,
$$\left(y^{2} - \frac{1}{2}\right)^{2} + \frac{19}{4}$$
/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\
| 4 ___ |atan\\/ 19 /| 4 ___ |atan\\/ 19 /|| | 4 ___ |atan\\/ 19 /| 4 ___ |atan\\/ 19 /|| | 4 ___ |atan\\/ 19 /| 4 ___ |atan\\/ 19 /|| | 4 ___ |atan\\/ 19 /| 4 ___ |atan\\/ 19 /||
|x + \/ 5 *cos|------------| + I*\/ 5 *sin|------------||*|x + \/ 5 *cos|------------| - I*\/ 5 *sin|------------||*|x + - \/ 5 *cos|------------| + I*\/ 5 *sin|------------||*|x + - \/ 5 *cos|------------| - I*\/ 5 *sin|------------||
\ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(x + \left(\sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right)$$
(((x + 5^(1/4)*cos(atan(sqrt(19))/2) + i*5^(1/4)*sin(atan(sqrt(19))/2))*(x + 5^(1/4)*cos(atan(sqrt(19))/2) - i*5^(1/4)*sin(atan(sqrt(19))/2)))*(x - 5^(1/4)*cos(atan(sqrt(19))/2) + i*5^(1/4)*sin(atan(sqrt(19))/2)))*(x - 5^(1/4)*cos(atan(sqrt(19))/2) - i*5^(1/4)*sin(atan(sqrt(19))/2))