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

Expresión a simplificar:

Solución

Ha introducido [src]
 4    2    
y  + y  + 4
(y4+y2)+4\left(y^{4} + y^{2}\right) + 4
y^4 + y^2 + 4
Factorización [src]
/             /    /  ____\\              /    /  ____\\\ /             /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\
|      ___    |atan\\/ 15 /|       ___    |atan\\/ 15 /|| |      ___    |atan\\/ 15 /|       ___    |atan\\/ 15 /|| |        ___    |atan\\/ 15 /|       ___    |atan\\/ 15 /|| |        ___    |atan\\/ 15 /|       ___    |atan\\/ 15 /||
|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+(2sin(atan(15)2)2icos(atan(15)2)))(x+(2sin(atan(15)2)+2icos(atan(15)2)))(x+(2sin(atan(15)2)+2icos(atan(15)2)))(x+(2sin(atan(15)2)2icos(atan(15)2)))\left(x + \left(\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} - \sqrt{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} + \sqrt{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} + \sqrt{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt{2} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)} - \sqrt{2} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}\right)\right)
(((x + sqrt(2)*sin(atan(sqrt(15))/2) + i*sqrt(2)*cos(atan(sqrt(15))/2))*(x + sqrt(2)*sin(atan(sqrt(15))/2) - i*sqrt(2)*cos(atan(sqrt(15))/2)))*(x - sqrt(2)*sin(atan(sqrt(15))/2) + i*sqrt(2)*cos(atan(sqrt(15))/2)))*(x - sqrt(2)*sin(atan(sqrt(15))/2) - i*sqrt(2)*cos(atan(sqrt(15))/2))
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(y4+y2)+4\left(y^{4} + y^{2}\right) + 4
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=4acb24an = \frac{4 a c - b^{2}}{4 a}
En nuestro caso
a=1a = 1
b=1b = 1
c=4c = 4
Entonces
m=12m = \frac{1}{2}
n=154n = \frac{15}{4}
Pues,
(y2+12)2+154\left(y^{2} + \frac{1}{2}\right)^{2} + \frac{15}{4}
Simplificación general [src]
     2    4
4 + y  + y 
y4+y2+4y^{4} + y^{2} + 4
4 + y^2 + y^4
Compilar la expresión [src]
     2    4
4 + y  + y 
y4+y2+4y^{4} + y^{2} + 4
4 + y^2 + y^4
Combinatoria [src]
     2    4
4 + y  + y 
y4+y2+4y^{4} + y^{2} + 4
4 + y^2 + y^4
Denominador común [src]
     2    4
4 + y  + y 
y4+y2+4y^{4} + y^{2} + 4
4 + y^2 + y^4
Respuesta numérica [src]
4.0 + y^2 + y^4
4.0 + y^2 + y^4
Parte trigonométrica [src]
     2    4
4 + y  + y 
y4+y2+4y^{4} + y^{2} + 4
4 + y^2 + y^4
Denominador racional [src]
     2    4
4 + y  + y 
y4+y2+4y^{4} + y^{2} + 4
4 + y^2 + y^4
Potencias [src]
     2    4
4 + y  + y 
y4+y2+4y^{4} + y^{2} + 4
4 + y^2 + y^4
Unión de expresiones racionales [src]
     2 /     2\
4 + y *\1 + y /
y2(y2+1)+4y^{2} \left(y^{2} + 1\right) + 4
4 + y^2*(1 + y^2)