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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
 4    2    
x  + x  + 5
(x4+x2)+5\left(x^{4} + x^{2}\right) + 5 
x^4 + x^2 + 5
Simplificación general [src] 
     2    4
5 + x  + x
x4+x2+5x^{4} + x^{2} + 5 
5 + x^2 + x^4
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(x4+x2)+5\left(x^{4} + x^{2}\right) + 5
Para eso usemos la fórmula
ax4+bx2+c=a(m+x2)2+na x^{4} + b x^{2} + c = a \left(m + x^{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,
(x2+12)2+194\left(x^{2} + \frac{1}{2}\right)^{2} + \frac{19}{4}
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))
Parte trigonométrica [src] 
     2    4
5 + x  + x
x4+x2+5x^{4} + x^{2} + 5 
5 + x^2 + x^4
Potencias [src] 
     2    4
5 + x  + x
x4+x2+5x^{4} + x^{2} + 5 
5 + x^2 + x^4
Denominador común [src] 
     2    4
5 + x  + x
x4+x2+5x^{4} + x^{2} + 5 
5 + x^2 + x^4
Denominador racional [src] 
     2    4
5 + x  + x
x4+x2+5x^{4} + x^{2} + 5 
5 + x^2 + x^4
Compilar la expresión [src] 
     2    4
5 + x  + x
x4+x2+5x^{4} + x^{2} + 5 
5 + x^2 + x^4
Respuesta numérica [src] 
5.0 + x^2 + x^4
5.0 + x^2 + x^4
Combinatoria [src] 
     2    4
5 + x  + x
x4+x2+5x^{4} + x^{2} + 5 
5 + x^2 + x^4
Unión de expresiones racionales [src] 
     2 /     2\
5 + x *\1 + x /
x2(x2+1)+5x^{2} \left(x^{2} + 1\right) + 5 
5 + x^2*(1 + x^2)
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