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Simplificación de expresiones/ Expresar el cuadrado perfecto/ -y^4+y^2-6

Descomponer -y^4+y^2-6 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src] 
   4    2    
- y  + y  - 6
(−y4+y2)−6\left(- y^{4} + y^{2}\right) - 6 
-y^4 + y^2 - 6
Simplificación general [src] 
      2    4
-6 + y  - y
−y4+y2−6- y^{4} + y^{2} - 6 
-6 + y^2 - y^4
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(−y4+y2)−6\left(- y^{4} + y^{2}\right) - 6
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=4ac−b24an = \frac{4 a c - b^{2}}{4 a}
En nuestro caso
a=−1a = -1
b=1b = 1
c=−6c = -6
Entonces
m=−12m = - \frac{1}{2}
n=−234n = - \frac{23}{4}
Pues,
−(y2−12)2−234- \left(y^{2} - \frac{1}{2}\right)^{2} - \frac{23}{4}
Factorización [src] 
/             /    /  ____\\              /    /  ____\\\ /             /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\ /               /    /  ____\\              /    /  ____\\\
|    4 ___    |atan\\/ 23 /|     4 ___    |atan\\/ 23 /|| |    4 ___    |atan\\/ 23 /|     4 ___    |atan\\/ 23 /|| |      4 ___    |atan\\/ 23 /|     4 ___    |atan\\/ 23 /|| |      4 ___    |atan\\/ 23 /|     4 ___    |atan\\/ 23 /||
|x + \/ 6 *cos|------------| + I*\/ 6 *sin|------------||*|x + \/ 6 *cos|------------| - I*\/ 6 *sin|------------||*|x + - \/ 6 *cos|------------| + I*\/ 6 *sin|------------||*|x + - \/ 6 *cos|------------| - I*\/ 6 *sin|------------||
\             \     2      /              \     2      // \             \     2      /              \     2      // \               \     2      /              \     2      // \               \     2      /              \     2      //
(x+(64cos⁡(atan⁡(23)2)−64isin⁡(atan⁡(23)2)))(x+(64cos⁡(atan⁡(23)2)+64isin⁡(atan⁡(23)2)))(x+(−64cos⁡(atan⁡(23)2)+64isin⁡(atan⁡(23)2)))(x+(−64cos⁡(atan⁡(23)2)−64isin⁡(atan⁡(23)2)))\left(x + \left(\sqrt[4]{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)} - \sqrt[4]{6} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)} + \sqrt[4]{6} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)} + \sqrt[4]{6} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{6} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)} - \sqrt[4]{6} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{23} \right)}}{2} \right)}\right)\right) 
(((x + 6^(1/4)*cos(atan(sqrt(23))/2) + i*6^(1/4)*sin(atan(sqrt(23))/2))*(x + 6^(1/4)*cos(atan(sqrt(23))/2) - i*6^(1/4)*sin(atan(sqrt(23))/2)))*(x - 6^(1/4)*cos(atan(sqrt(23))/2) + i*6^(1/4)*sin(atan(sqrt(23))/2)))*(x - 6^(1/4)*cos(atan(sqrt(23))/2) - i*6^(1/4)*sin(atan(sqrt(23))/2))
Denominador racional [src] 
      2    4
-6 + y  - y
−y4+y2−6- y^{4} + y^{2} - 6 
-6 + y^2 - y^4
Combinatoria [src] 
      2    4
-6 + y  - y
−y4+y2−6- y^{4} + y^{2} - 6 
-6 + y^2 - y^4
Compilar la expresión [src] 
      2    4
-6 + y  - y
−y4+y2−6- y^{4} + y^{2} - 6 
-6 + y^2 - y^4
Unión de expresiones racionales [src] 
      2 /     2\
-6 + y *\1 - y /
y2(1−y2)−6y^{2} \left(1 - y^{2}\right) - 6 
-6 + y^2*(1 - y^2)
Respuesta numérica [src] 
-6.0 + y^2 - y^4
-6.0 + y^2 - y^4
Denominador común [src] 
      2    4
-6 + y  - y
−y4+y2−6- y^{4} + y^{2} - 6 
-6 + y^2 - y^4
Potencias [src] 
      2    4
-6 + y  - y
−y4+y2−6- y^{4} + y^{2} - 6 
-6 + y^2 - y^4
Parte trigonométrica [src] 
      2    4
-6 + y  - y
−y4+y2−6- y^{4} + y^{2} - 6 
-6 + y^2 - y^4
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