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Simplificación de expresiones/ Expresar el cuadrado perfecto/ 5*x^4+3*x^2+2

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   4      2    
5*x  + 3*x  + 2
(5x4+3x2)+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
5x4+3x2+25 x^{4} + 3 x^{2} + 2 
2 + 3*x^2 + 5*x^4
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(5x4+3x2)+2\left(5 x^{4} + 3 x^{2}\right) + 2
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=5a = 5
b=3b = 3
c=2c = 2
Entonces
m=310m = \frac{3}{10}
n=3120n = \frac{31}{20}
Pues,
5(x2+310)2+31205 \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   *sin|------------|   I*\/ 2 *5   *cos|------------|| |    \/ 2 *5   *sin|------------|   I*\/ 2 *5   *cos|------------|| |      \/ 2 *5   *sin|------------|   I*\/ 2 *5   *cos|------------|| |      \/ 2 *5   *sin|------------|   I*\/ 2 *5   *cos|------------||
|                  \     2      /                   \     2      /| |                  \     2      /                   \     2      /| |                    \     2      /                   \     2      /| |                    \     2      /                   \     2      /|
|x + ---------------------------- + ------------------------------|*|x + ---------------------------- - ------------------------------|*|x + - ---------------------------- + ------------------------------|*|x + - ---------------------------- - ------------------------------|
\                 5                               5               / \                 5                               5               / \                   5                               5               / \                   5                               5               /
(x+(24⋅534sin⁡(atan⁡(313)2)5−24⋅534icos⁡(atan⁡(313)2)5))(x+(24⋅534sin⁡(atan⁡(313)2)5+24⋅534icos⁡(atan⁡(313)2)5))(x+(−24⋅534sin⁡(atan⁡(313)2)5+24⋅534icos⁡(atan⁡(313)2)5))(x+(−24⋅534sin⁡(atan⁡(313)2)5−24⋅534icos⁡(atan⁡(313)2)5))\left(x + \left(\frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\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}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\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}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} + \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\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}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5} - \frac{\sqrt[4]{2} \cdot 5^{\frac{3}{4}} i \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{31}}{3} \right)}}{2} \right)}}{5}\right)\right) 
(((x + 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 + i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5)*(x + 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 - i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 + i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5))*(x - 2^(1/4)*5^(3/4)*sin(atan(sqrt(31)/3)/2)/5 - i*2^(1/4)*5^(3/4)*cos(atan(sqrt(31)/3)/2)/5)
Compilar la expresión [src] 
       2      4
2 + 3*x  + 5*x
5x4+3x2+25 x^{4} + 3 x^{2} + 2 
2 + 3*x^2 + 5*x^4
Parte trigonométrica [src] 
       2      4
2 + 3*x  + 5*x
5x4+3x2+25 x^{4} + 3 x^{2} + 2 
2 + 3*x^2 + 5*x^4
Denominador racional [src] 
       2      4
2 + 3*x  + 5*x
5x4+3x2+25 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 /
x2(5x2+3)+2x^{2} \left(5 x^{2} + 3\right) + 2 
2 + x^2*(3 + 5*x^2)
Combinatoria [src] 
       2      4
2 + 3*x  + 5*x
5x4+3x2+25 x^{4} + 3 x^{2} + 2 
2 + 3*x^2 + 5*x^4
Respuesta numérica [src] 
2.0 + 3.0*x^2 + 5.0*x^4
2.0 + 3.0*x^2 + 5.0*x^4
Potencias [src] 
       2      4
2 + 3*x  + 5*x
5x4+3x2+25 x^{4} + 3 x^{2} + 2 
2 + 3*x^2 + 5*x^4
Denominador común [src] 
       2      4
2 + 3*x  + 5*x
5x4+3x2+25 x^{4} + 3 x^{2} + 2 
2 + 3*x^2 + 5*x^4
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