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Descomponer y^4-y^2+7 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src]
 4    2    
y  - y  + 7
(y4y2)+7\left(y^{4} - y^{2}\right) + 7
y^4 - y^2 + 7
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(y4y2)+7\left(y^{4} - y^{2}\right) + 7
Para eso usemos la fórmula
ay4+by2+c=a(m+y2)2+na y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n
donde
m=b2am = \frac{b}{2 a}
n=4acb24an = \frac{4 a c - b^{2}}{4 a}
En nuestro caso
a=1a = 1
b=1b = -1
c=7c = 7
Entonces
m=12m = - \frac{1}{2}
n=274n = \frac{27}{4}
Pues,
(y212)2+274\left(y^{2} - \frac{1}{2}\right)^{2} + \frac{27}{4}
Simplificación general [src]
     4    2
7 + y  - y 
y4y2+7y^{4} - y^{2} + 7
7 + y^4 - y^2
Factorización [src]
/             /    /    ___\\              /    /    ___\\\ /             /    /    ___\\              /    /    ___\\\ /               /    /    ___\\              /    /    ___\\\ /               /    /    ___\\              /    /    ___\\\
|    4 ___    |atan\3*\/ 3 /|     4 ___    |atan\3*\/ 3 /|| |    4 ___    |atan\3*\/ 3 /|     4 ___    |atan\3*\/ 3 /|| |      4 ___    |atan\3*\/ 3 /|     4 ___    |atan\3*\/ 3 /|| |      4 ___    |atan\3*\/ 3 /|     4 ___    |atan\3*\/ 3 /||
|x + \/ 7 *cos|-------------| + I*\/ 7 *sin|-------------||*|x + \/ 7 *cos|-------------| - I*\/ 7 *sin|-------------||*|x + - \/ 7 *cos|-------------| + I*\/ 7 *sin|-------------||*|x + - \/ 7 *cos|-------------| - I*\/ 7 *sin|-------------||
\             \      2      /              \      2      // \             \      2      /              \      2      // \               \      2      /              \      2      // \               \      2      /              \      2      //
(x+(74cos(atan(33)2)74isin(atan(33)2)))(x+(74cos(atan(33)2)+74isin(atan(33)2)))(x+(74cos(atan(33)2)+74isin(atan(33)2)))(x+(74cos(atan(33)2)74isin(atan(33)2)))\left(x + \left(\sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} - \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} + \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} + \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{7} \cos{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)} - \sqrt[4]{7} i \sin{\left(\frac{\operatorname{atan}{\left(3 \sqrt{3} \right)}}{2} \right)}\right)\right)
(((x + 7^(1/4)*cos(atan(3*sqrt(3))/2) + i*7^(1/4)*sin(atan(3*sqrt(3))/2))*(x + 7^(1/4)*cos(atan(3*sqrt(3))/2) - i*7^(1/4)*sin(atan(3*sqrt(3))/2)))*(x - 7^(1/4)*cos(atan(3*sqrt(3))/2) + i*7^(1/4)*sin(atan(3*sqrt(3))/2)))*(x - 7^(1/4)*cos(atan(3*sqrt(3))/2) - i*7^(1/4)*sin(atan(3*sqrt(3))/2))
Denominador común [src]
     4    2
7 + y  - y 
y4y2+7y^{4} - y^{2} + 7
7 + y^4 - y^2
Combinatoria [src]
     4    2
7 + y  - y 
y4y2+7y^{4} - y^{2} + 7
7 + y^4 - y^2
Respuesta numérica [src]
7.0 + y^4 - y^2
7.0 + y^4 - y^2
Compilar la expresión [src]
     4    2
7 + y  - y 
y4y2+7y^{4} - y^{2} + 7
7 + y^4 - y^2
Potencias [src]
     4    2
7 + y  - y 
y4y2+7y^{4} - y^{2} + 7
7 + y^4 - y^2
Parte trigonométrica [src]
     4    2
7 + y  - y 
y4y2+7y^{4} - y^{2} + 7
7 + y^4 - y^2
Denominador racional [src]
     4    2
7 + y  - y 
y4y2+7y^{4} - y^{2} + 7
7 + y^4 - y^2
Unión de expresiones racionales [src]
     2 /      2\
7 + y *\-1 + y /
y2(y21)+7y^{2} \left(y^{2} - 1\right) + 7
7 + y^2*(-1 + y^2)