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Simplificación de expresiones/ Expresar el cuadrado perfecto/ y^4+y^2+5

Descomponer y^4+y^2+5 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src] 
 4    2    
y  + y  + 5
(y4+y2)+5\left(y^{4} + y^{2}\right) + 5 
y^4 + y^2 + 5
Factorización [src] 
/             /    /  ____\\              /    /  ____\\\ /             /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\
|    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 *sin|------------| + I*\/ 5 *cos|------------||*|x + \/ 5 *sin|------------| - I*\/ 5 *cos|------------||*|x + - \/ 5 *sin|------------| + I*\/ 5 *cos|------------||*|x + - \/ 5 *sin|------------| - I*\/ 5 *cos|------------||
\             \     2      /              \     2      // \             \     2      /              \     2      // \               \     2      /              \     2      // \               \     2      /              \     2      //
(x+(54sin⁡(atan⁡(19)2)−54icos⁡(atan⁡(19)2)))(x+(54sin⁡(atan⁡(19)2)+54icos⁡(atan⁡(19)2)))(x+(−54sin⁡(atan⁡(19)2)+54icos⁡(atan⁡(19)2)))(x+(−54sin⁡(atan⁡(19)2)−54icos⁡(atan⁡(19)2)))\left(x + \left(\sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) 
(((x + 5^(1/4)*sin(atan(sqrt(19))/2) + i*5^(1/4)*cos(atan(sqrt(19))/2))*(x + 5^(1/4)*sin(atan(sqrt(19))/2) - i*5^(1/4)*cos(atan(sqrt(19))/2)))*(x - 5^(1/4)*sin(atan(sqrt(19))/2) + i*5^(1/4)*cos(atan(sqrt(19))/2)))*(x - 5^(1/4)*sin(atan(sqrt(19))/2) - i*5^(1/4)*cos(atan(sqrt(19))/2))
Simplificación general [src] 
     2    4
5 + y  + y
y4+y2+5y^{4} + y^{2} + 5 
5 + y^2 + y^4
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(y4+y2)+5\left(y^{4} + y^{2}\right) + 5
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=5c = 5
Entonces
m=12m = \frac{1}{2}
n=194n = \frac{19}{4}
Pues,
(y2+12)2+194\left(y^{2} + \frac{1}{2}\right)^{2} + \frac{19}{4}
Denominador racional [src] 
     2    4
5 + y  + y
y4+y2+5y^{4} + y^{2} + 5 
5 + y^2 + y^4
Denominador común [src] 
     2    4
5 + y  + y
y4+y2+5y^{4} + y^{2} + 5 
5 + y^2 + y^4
Parte trigonométrica [src] 
     2    4
5 + y  + y
y4+y2+5y^{4} + y^{2} + 5 
5 + y^2 + y^4
Compilar la expresión [src] 
     2    4
5 + y  + y
y4+y2+5y^{4} + y^{2} + 5 
5 + y^2 + y^4
Potencias [src] 
     2    4
5 + y  + y
y4+y2+5y^{4} + y^{2} + 5 
5 + y^2 + y^4
Respuesta numérica [src] 
5.0 + y^2 + y^4
5.0 + y^2 + y^4
Unión de expresiones racionales [src] 
     2 /     2\
5 + y *\1 + y /
y2(y2+1)+5y^{2} \left(y^{2} + 1\right) + 5 
5 + y^2*(1 + y^2)
Combinatoria [src] 
     2    4
5 + y  + y
y4+y2+5y^{4} + y^{2} + 5 
5 + y^2 + y^4
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