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

Expresión a simplificar:

Solución

Ha introducido [src]
 4    2    
x  + x  + 5
$$\left(x^{4} + x^{2}\right) + 5$$
x^4 + x^2 + 5
Simplificación general [src]
     2    4
5 + x  + x 
$$x^{4} + x^{2} + 5$$
5 + x^2 + x^4
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(x^{4} + x^{2}\right) + 5$$
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
$$a = 1$$
$$b = 1$$
$$c = 5$$
Entonces
$$m = \frac{1}{2}$$
$$n = \frac{19}{4}$$
Pues,
$$\left(x^{2} + \frac{1}{2}\right)^{2} + \frac{19}{4}$$
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      //
$$\left(x + \left(\sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} + \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{5} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)} - \sqrt[4]{5} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{19} \right)}}{2} \right)}\right)\right)$$
(((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 
$$x^{4} + x^{2} + 5$$
5 + x^2 + x^4
Potencias [src]
     2    4
5 + x  + x 
$$x^{4} + x^{2} + 5$$
5 + x^2 + x^4
Denominador común [src]
     2    4
5 + x  + x 
$$x^{4} + x^{2} + 5$$
5 + x^2 + x^4
Denominador racional [src]
     2    4
5 + x  + x 
$$x^{4} + x^{2} + 5$$
5 + x^2 + x^4
Compilar la expresión [src]
     2    4
5 + x  + x 
$$x^{4} + x^{2} + 5$$
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 
$$x^{4} + x^{2} + 5$$
5 + x^2 + x^4
Unión de expresiones racionales [src]
     2 /     2\
5 + x *\1 + x /
$$x^{2} \left(x^{2} + 1\right) + 5$$
5 + x^2*(1 + x^2)