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Simplificación de expresiones/ Expresar el cuadrado perfecto/ -y^4-y^2-8

Descomponer -y^4-y^2-8 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src] 
   4    2    
- y  - y  - 8
(−y4−y2)−8\left(- y^{4} - y^{2}\right) - 8 
-y^4 - y^2 - 8
Factorización [src] 
/            /    /  ____\\             /    /  ____\\\ /            /    /  ____\\             /    /  ____\\\ /              /    /  ____\\             /    /  ____\\\ /              /    /  ____\\             /    /  ____\\\
|     3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /|| |     3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /|| |       3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /|| |       3/4    |atan\\/ 31 /|      3/4    |atan\\/ 31 /||
|x + 2   *sin|------------| + I*2   *cos|------------||*|x + 2   *sin|------------| - I*2   *cos|------------||*|x + - 2   *sin|------------| + I*2   *cos|------------||*|x + - 2   *sin|------------| - I*2   *cos|------------||
\            \     2      /             \     2      // \            \     2      /             \     2      // \              \     2      /             \     2      // \              \     2      /             \     2      //
(x+(234sin⁡(atan⁡(31)2)−234icos⁡(atan⁡(31)2)))(x+(234sin⁡(atan⁡(31)2)+234icos⁡(atan⁡(31)2)))(x+(−234sin⁡(atan⁡(31)2)+234icos⁡(atan⁡(31)2)))(x+(−234sin⁡(atan⁡(31)2)−234icos⁡(atan⁡(31)2)))\left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right) \left(x + \left(2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} + 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right) \left(x + \left(- 2^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)} - 2^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{31} \right)}}{2} \right)}\right)\right) 
(((x + 2^(3/4)*sin(atan(sqrt(31))/2) + i*2^(3/4)*cos(atan(sqrt(31))/2))*(x + 2^(3/4)*sin(atan(sqrt(31))/2) - i*2^(3/4)*cos(atan(sqrt(31))/2)))*(x - 2^(3/4)*sin(atan(sqrt(31))/2) + i*2^(3/4)*cos(atan(sqrt(31))/2)))*(x - 2^(3/4)*sin(atan(sqrt(31))/2) - i*2^(3/4)*cos(atan(sqrt(31))/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(−y4−y2)−8\left(- y^{4} - y^{2}\right) - 8
Para eso usemos la fórmula
ay4+by2+c=a(m+y2)2+na y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n
donde
m=b2am = \frac{b}{2 a}
n=4ac−b24an = \frac{4 a c - b^{2}}{4 a}
En nuestro caso
a=−1a = -1
b=−1b = -1
c=−8c = -8
Entonces
m=12m = \frac{1}{2}
n=−314n = - \frac{31}{4}
Pues,
−(y2+12)2−314- \left(y^{2} + \frac{1}{2}\right)^{2} - \frac{31}{4}
Simplificación general [src] 
      2    4
-8 - y  - y
−y4−y2−8- y^{4} - y^{2} - 8 
-8 - y^2 - y^4
Respuesta numérica [src] 
-8.0 - y^2 - y^4
-8.0 - y^2 - y^4
Parte trigonométrica [src] 
      2    4
-8 - y  - y
−y4−y2−8- y^{4} - y^{2} - 8 
-8 - y^2 - y^4
Denominador racional [src] 
      2    4
-8 - y  - y
−y4−y2−8- y^{4} - y^{2} - 8 
-8 - y^2 - y^4
Denominador común [src] 
      2    4
-8 - y  - y
−y4−y2−8- y^{4} - y^{2} - 8 
-8 - y^2 - y^4
Unión de expresiones racionales [src] 
      2 /      2\
-8 + y *\-1 - y /
y2(−y2−1)−8y^{2} \left(- y^{2} - 1\right) - 8 
-8 + y^2*(-1 - y^2)
Combinatoria [src] 
      2    4
-8 - y  - y
−y4−y2−8- y^{4} - y^{2} - 8 
-8 - y^2 - y^4
Compilar la expresión [src] 
      2    4
-8 - y  - y
−y4−y2−8- y^{4} - y^{2} - 8 
-8 - y^2 - y^4
Potencias [src] 
      2    4
-8 - y  - y
−y4−y2−8- y^{4} - y^{2} - 8 
-8 - y^2 - y^4
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