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Descomponer 5*x^4-3*x^2+2 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src]
   4      2    
5*x  - 3*x  + 2
$$\left(5 x^{4} - 3 x^{2}\right) + 2$$
5*x^4 - 3*x^2 + 2
Simplificación general [src]
       2      4
2 - 3*x  + 5*x 
$$5 x^{4} - 3 x^{2} + 2$$
2 - 3*x^2 + 5*x^4
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(5 x^{4} - 3 x^{2}\right) + 2$$
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
$$a = 5$$
$$b = -3$$
$$c = 2$$
Entonces
$$m = - \frac{3}{10}$$
$$n = \frac{31}{20}$$
Pues,
$$5 \left(x^{2} - \frac{3}{10}\right)^{2} + \frac{31}{20}$$
Factorización [src]
/                  /    /  ____\\                   /    /  ____\\\ /                  /    /  ____\\                   /    /  ____\\\ /                    /    /  ____\\                   /    /  ____\\\ /                    /    /  ____\\                   /    /  ____\\\
|                  |    |\/ 31 ||                   |    |\/ 31 ||| |                  |    |\/ 31 ||                   |    |\/ 31 ||| |                    |    |\/ 31 ||                   |    |\/ 31 ||| |                    |    |\/ 31 ||                   |    |\/ 31 |||
|                  |atan|------||                   |atan|------||| |                  |atan|------||                   |atan|------||| |                    |atan|------||                   |atan|------||| |                    |atan|------||                   |atan|------|||
|    4 ___  3/4    |    \  3   /|     4 ___  3/4    |    \  3   /|| |    4 ___  3/4    |    \  3   /|     4 ___  3/4    |    \  3   /|| |      4 ___  3/4    |    \  3   /|     4 ___  3/4    |    \  3   /|| |      4 ___  3/4    |    \  3   /|     4 ___  3/4    |    \  3   /||
|    \/ 2 *5   *cos|------------|   I*\/ 2 *5   *sin|------------|| |    \/ 2 *5   *cos|------------|   I*\/ 2 *5   *sin|------------|| |      \/ 2 *5   *cos|------------|   I*\/ 2 *5   *sin|------------|| |      \/ 2 *5   *cos|------------|   I*\/ 2 *5   *sin|------------||
|                  \     2      /                   \     2      /| |                  \     2      /                   \     2      /| |                    \     2      /                   \     2      /| |                    \     2      /                   \     2      /|
|x + ---------------------------- + ------------------------------|*|x + ---------------------------- - ------------------------------|*|x + - ---------------------------- + ------------------------------|*|x + - ---------------------------- - ------------------------------|
\                 5                               5               / \                 5                               5               / \                   5                               5               / \                   5                               5               /
$$\left(x + \left(\frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(\frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(- \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right) \left(x + \left(- \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right)$$
(((x + 2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5 + i*2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5)*(x + 2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5 - i*2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5 + i*2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5 - i*2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5)
Compilar la expresión [src]
       2      4
2 - 3*x  + 5*x 
$$5 x^{4} - 3 x^{2} + 2$$
2 - 3*x^2 + 5*x^4
Respuesta numérica [src]
2.0 + 5.0*x^4 - 3.0*x^2
2.0 + 5.0*x^4 - 3.0*x^2
Parte trigonométrica [src]
       2      4
2 - 3*x  + 5*x 
$$5 x^{4} - 3 x^{2} + 2$$
2 - 3*x^2 + 5*x^4
Denominador común [src]
       2      4
2 - 3*x  + 5*x 
$$5 x^{4} - 3 x^{2} + 2$$
2 - 3*x^2 + 5*x^4
Unión de expresiones racionales [src]
     2 /        2\
2 + x *\-3 + 5*x /
$$x^{2} \left(5 x^{2} - 3\right) + 2$$
2 + x^2*(-3 + 5*x^2)
Combinatoria [src]
       2      4
2 - 3*x  + 5*x 
$$5 x^{4} - 3 x^{2} + 2$$
2 - 3*x^2 + 5*x^4
Denominador racional [src]
       2      4
2 - 3*x  + 5*x 
$$5 x^{4} - 3 x^{2} + 2$$
2 - 3*x^2 + 5*x^4
Potencias [src]
       2      4
2 - 3*x  + 5*x 
$$5 x^{4} - 3 x^{2} + 2$$
2 - 3*x^2 + 5*x^4