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Simplificación de expresiones/ Expresar el cuadrado perfecto/ -y^4+y^2-10

Descomponer -y^4+y^2-10 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src] 
   4    2     
- y  + y  - 10
(−y4+y2)−10\left(- y^{4} + y^{2}\right) - 10 
-y^4 + y^2 - 10
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(−y4+y2)−10\left(- y^{4} + y^{2}\right) - 10
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=−10c = -10
Entonces
m=−12m = - \frac{1}{2}
n=−394n = - \frac{39}{4}
Pues,
−(y2−12)2−394- \left(y^{2} - \frac{1}{2}\right)^{2} - \frac{39}{4}
Simplificación general [src] 
       2    4
-10 + y  - y
−y4+y2−10- y^{4} + y^{2} - 10 
-10 + y^2 - y^4
Factorización [src] 
/              /    /  ____\\               /    /  ____\\\ /              /    /  ____\\               /    /  ____\\\ /                /    /  ____\\               /    /  ____\\\ /                /    /  ____\\               /    /  ____\\\
|    4 ____    |atan\\/ 39 /|     4 ____    |atan\\/ 39 /|| |    4 ____    |atan\\/ 39 /|     4 ____    |atan\\/ 39 /|| |      4 ____    |atan\\/ 39 /|     4 ____    |atan\\/ 39 /|| |      4 ____    |atan\\/ 39 /|     4 ____    |atan\\/ 39 /||
|x + \/ 10 *cos|------------| + I*\/ 10 *sin|------------||*|x + \/ 10 *cos|------------| - I*\/ 10 *sin|------------||*|x + - \/ 10 *cos|------------| + I*\/ 10 *sin|------------||*|x + - \/ 10 *cos|------------| - I*\/ 10 *sin|------------||
\              \     2      /               \     2      // \              \     2      /               \     2      // \                \     2      /               \     2      // \                \     2      /               \     2      //
(x+(104cos⁡(atan⁡(39)2)−104isin⁡(atan⁡(39)2)))(x+(104cos⁡(atan⁡(39)2)+104isin⁡(atan⁡(39)2)))(x+(−104cos⁡(atan⁡(39)2)+104isin⁡(atan⁡(39)2)))(x+(−104cos⁡(atan⁡(39)2)−104isin⁡(atan⁡(39)2)))\left(x + \left(\sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)} - \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)} + \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)} + \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{10} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)} - \sqrt[4]{10} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{39} \right)}}{2} \right)}\right)\right) 
(((x + 10^(1/4)*cos(atan(sqrt(39))/2) + i*10^(1/4)*sin(atan(sqrt(39))/2))*(x + 10^(1/4)*cos(atan(sqrt(39))/2) - i*10^(1/4)*sin(atan(sqrt(39))/2)))*(x - 10^(1/4)*cos(atan(sqrt(39))/2) + i*10^(1/4)*sin(atan(sqrt(39))/2)))*(x - 10^(1/4)*cos(atan(sqrt(39))/2) - i*10^(1/4)*sin(atan(sqrt(39))/2))
Potencias [src] 
       2    4
-10 + y  - y
−y4+y2−10- y^{4} + y^{2} - 10 
-10 + y^2 - y^4
Unión de expresiones racionales [src] 
       2 /     2\
-10 + y *\1 - y /
y2(1−y2)−10y^{2} \left(1 - y^{2}\right) - 10 
-10 + y^2*(1 - y^2)
Denominador común [src] 
       2    4
-10 + y  - y
−y4+y2−10- y^{4} + y^{2} - 10 
-10 + y^2 - y^4
Respuesta numérica [src] 
-10.0 + y^2 - y^4
-10.0 + y^2 - y^4
Denominador racional [src] 
       2    4
-10 + y  - y
−y4+y2−10- y^{4} + y^{2} - 10 
-10 + y^2 - y^4
Parte trigonométrica [src] 
       2    4
-10 + y  - y
−y4+y2−10- y^{4} + y^{2} - 10 
-10 + y^2 - y^4
Compilar la expresión [src] 
       2    4
-10 + y  - y
−y4+y2−10- y^{4} + y^{2} - 10 
-10 + y^2 - y^4
Combinatoria [src] 
       2    4
-10 + y  - y
−y4+y2−10- y^{4} + y^{2} - 10 
-10 + y^2 - y^4
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