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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
$$\left(x^{4} + 2 x^{2}\right) + 3$$ 
x^4 + 2*x^2 + 3
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(x^{4} + 2 x^{2}\right) + 3$$
Para eso usemos la fórmula
$$a x^{4} + b x^{2} + c = a \left(m + x^{2}\right)^{2} + n$$
donde
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
En nuestro caso
$$a = 1$$
$$b = 2$$
$$c = 3$$
Entonces
$$m = 1$$
$$n = 2$$
Pues,
$$\left(x^{2} + 1\right)^{2} + 2$$
Simplificación general [src] 
     4      2
3 + x  + 2*x
$$x^{4} + 2 x^{2} + 3$$ 
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     //
$$\left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right)$$ 
(((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
$$x^{4} + 2 x^{2} + 3$$ 
3 + x^4 + 2*x^2
Denominador común [src] 
     4      2
3 + x  + 2*x
$$x^{4} + 2 x^{2} + 3$$ 
3 + x^4 + 2*x^2
Denominador racional [src] 
     4      2
3 + x  + 2*x
$$x^{4} + 2 x^{2} + 3$$ 
3 + x^4 + 2*x^2
Potencias [src] 
     4      2
3 + x  + 2*x
$$x^{4} + 2 x^{2} + 3$$ 
3 + x^4 + 2*x^2
Combinatoria [src] 
     4      2
3 + x  + 2*x
$$x^{4} + 2 x^{2} + 3$$ 
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
$$x^{4} + 2 x^{2} + 3$$ 
3 + x^4 + 2*x^2
Unión de expresiones racionales [src] 
     2 /     2\
3 + x *\2 + x /
$$x^{2} \left(x^{2} + 2\right) + 3$$ 
3 + x^2*(2 + x^2)
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