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Simplificación de expresiones/ Expresar el cuadrado perfecto/ x^4+2*x^2+3

Descomponer x^4+2*x^2+3 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src] 
 4      2    
x  + 2*x  + 3
(x4+2x2)+3\left(x^{4} + 2 x^{2}\right) + 3 
x^4 + 2*x^2 + 3
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(x4+2x2)+3\left(x^{4} + 2 x^{2}\right) + 3
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=2b = 2
c=3c = 3
Entonces
m=1m = 1
n=2n = 2
Pues,
(x2+1)2+2\left(x^{2} + 1\right)^{2} + 2
Simplificación general [src] 
     4      2
3 + x  + 2*x
x4+2x2+3x^{4} + 2 x^{2} + 3 
3 + x^4 + 2*x^2
Factorización [src] 
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|    4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /|| |    4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /|| |      4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /|| |      4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /||
|x + \/ 3 *sin|-----------| + I*\/ 3 *cos|-----------||*|x + \/ 3 *sin|-----------| - I*\/ 3 *cos|-----------||*|x + - \/ 3 *sin|-----------| + I*\/ 3 *cos|-----------||*|x + - \/ 3 *sin|-----------| - I*\/ 3 *cos|-----------||
\             \     2     /              \     2     // \             \     2     /              \     2     // \               \     2     /              \     2     // \               \     2     /              \     2     //
(x+(34sin⁡(atan⁡(2)2)−34icos⁡(atan⁡(2)2)))(x+(34sin⁡(atan⁡(2)2)+34icos⁡(atan⁡(2)2)))(x+(−34sin⁡(atan⁡(2)2)+34icos⁡(atan⁡(2)2)))(x+(−34sin⁡(atan⁡(2)2)−34icos⁡(atan⁡(2)2)))\left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) 
(((x + 3^(1/4)*sin(atan(sqrt(2))/2) + i*3^(1/4)*cos(atan(sqrt(2))/2))*(x + 3^(1/4)*sin(atan(sqrt(2))/2) - i*3^(1/4)*cos(atan(sqrt(2))/2)))*(x - 3^(1/4)*sin(atan(sqrt(2))/2) + i*3^(1/4)*cos(atan(sqrt(2))/2)))*(x - 3^(1/4)*sin(atan(sqrt(2))/2) - i*3^(1/4)*cos(atan(sqrt(2))/2))
Parte trigonométrica [src] 
     4      2
3 + x  + 2*x
x4+2x2+3x^{4} + 2 x^{2} + 3 
3 + x^4 + 2*x^2
Denominador común [src] 
     4      2
3 + x  + 2*x
x4+2x2+3x^{4} + 2 x^{2} + 3 
3 + x^4 + 2*x^2
Denominador racional [src] 
     4      2
3 + x  + 2*x
x4+2x2+3x^{4} + 2 x^{2} + 3 
3 + x^4 + 2*x^2
Potencias [src] 
     4      2
3 + x  + 2*x
x4+2x2+3x^{4} + 2 x^{2} + 3 
3 + x^4 + 2*x^2
Combinatoria [src] 
     4      2
3 + x  + 2*x
x4+2x2+3x^{4} + 2 x^{2} + 3 
3 + x^4 + 2*x^2
Respuesta numérica [src] 
3.0 + x^4 + 2.0*x^2
3.0 + x^4 + 2.0*x^2
Compilar la expresión [src] 
     4      2
3 + x  + 2*x
x4+2x2+3x^{4} + 2 x^{2} + 3 
3 + x^4 + 2*x^2
Unión de expresiones racionales [src] 
     2 /     2\
3 + x *\2 + x /
x2(x2+2)+3x^{2} \left(x^{2} + 2\right) + 3 
3 + x^2*(2 + x^2)
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