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Descomponer 2*x^4-x^2+1 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src]
   4    2    
2*x  - x  + 1
$$\left(2 x^{4} - x^{2}\right) + 1$$
2*x^4 - x^2 + 1
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
$$\left(2 x^{4} - x^{2}\right) + 1$$
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
$$a = 2$$
$$b = -1$$
$$c = 1$$
Entonces
$$m = - \frac{1}{4}$$
$$n = \frac{7}{8}$$
Pues,
$$2 \left(x^{2} - \frac{1}{4}\right)^{2} + \frac{7}{8}$$
Simplificación general [src]
     2      4
1 - x  + 2*x 
$$2 x^{4} - x^{2} + 1$$
1 - x^2 + 2*x^4
Factorización [src]
/            /    /  ___\\             /    /  ___\\\ /            /    /  ___\\             /    /  ___\\\ /              /    /  ___\\             /    /  ___\\\ /              /    /  ___\\             /    /  ___\\\
|     3/4    |atan\\/ 7 /|      3/4    |atan\\/ 7 /|| |     3/4    |atan\\/ 7 /|      3/4    |atan\\/ 7 /|| |       3/4    |atan\\/ 7 /|      3/4    |atan\\/ 7 /|| |       3/4    |atan\\/ 7 /|      3/4    |atan\\/ 7 /||
|    2   *cos|-----------|   I*2   *sin|-----------|| |    2   *cos|-----------|   I*2   *sin|-----------|| |      2   *cos|-----------|   I*2   *sin|-----------|| |      2   *cos|-----------|   I*2   *sin|-----------||
|            \     2     /             \     2     /| |            \     2     /             \     2     /| |              \     2     /             \     2     /| |              \     2     /             \     2     /|
|x + --------------------- + -----------------------|*|x + --------------------- - -----------------------|*|x + - --------------------- + -----------------------|*|x + - --------------------- - -----------------------|
\              2                        2           / \              2                        2           / \                2                        2           / \                2                        2           /
$$\left(x + \left(\frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(\frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(- \frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right)\right) \left(x + \left(- \frac{2^{\frac{3}{4}} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{2^{\frac{3}{4}} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right)\right)$$
(((x + 2^(3/4)*cos(atan(sqrt(7))/2)/2 + i*2^(3/4)*sin(atan(sqrt(7))/2)/2)*(x + 2^(3/4)*cos(atan(sqrt(7))/2)/2 - i*2^(3/4)*sin(atan(sqrt(7))/2)/2))*(x - 2^(3/4)*cos(atan(sqrt(7))/2)/2 + i*2^(3/4)*sin(atan(sqrt(7))/2)/2))*(x - 2^(3/4)*cos(atan(sqrt(7))/2)/2 - i*2^(3/4)*sin(atan(sqrt(7))/2)/2)
Respuesta numérica [src]
1.0 - x^2 + 2.0*x^4
1.0 - x^2 + 2.0*x^4
Denominador racional [src]
     2      4
1 - x  + 2*x 
$$2 x^{4} - x^{2} + 1$$
1 - x^2 + 2*x^4
Compilar la expresión [src]
     2      4
1 - x  + 2*x 
$$2 x^{4} - x^{2} + 1$$
1 - x^2 + 2*x^4
Potencias [src]
     2      4
1 - x  + 2*x 
$$2 x^{4} - x^{2} + 1$$
1 - x^2 + 2*x^4
Denominador común [src]
     2      4
1 - x  + 2*x 
$$2 x^{4} - x^{2} + 1$$
1 - x^2 + 2*x^4
Parte trigonométrica [src]
     2      4
1 - x  + 2*x 
$$2 x^{4} - x^{2} + 1$$
1 - x^2 + 2*x^4
Combinatoria [src]
     2      4
1 - x  + 2*x 
$$2 x^{4} - x^{2} + 1$$
1 - x^2 + 2*x^4
Unión de expresiones racionales [src]
     2 /        2\
1 + x *\-1 + 2*x /
$$x^{2} \left(2 x^{2} - 1\right) + 1$$
1 + x^2*(-1 + 2*x^2)