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Simplificación de expresiones/ Expresar el cuadrado perfecto/ y^4-5*y^2+15

Descomponer y^4-5*y^2+15 al cuadrado

Expresión a simplificar:

Solución

Ha introducido [src] 
 4      2     
y  - 5*y  + 15
(y4−5y2)+15\left(y^{4} - 5 y^{2}\right) + 15 
y^4 - 5*y^2 + 15
Expresión del cuadrado perfecto
Expresemos el cuadrado perfecto del trinomio cuadrático
(y4−5y2)+15\left(y^{4} - 5 y^{2}\right) + 15
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=−5b = -5
c=15c = 15
Entonces
m=−52m = - \frac{5}{2}
n=354n = \frac{35}{4}
Pues,
(y2−52)2+354\left(y^{2} - \frac{5}{2}\right)^{2} + \frac{35}{4}
Simplificación general [src] 
      4      2
15 + y  - 5*y
y4−5y2+15y^{4} - 5 y^{2} + 15 
15 + y^4 - 5*y^2
Factorización [src] 
/              /    /  ____\\               /    /  ____\\\ /              /    /  ____\\               /    /  ____\\\ /                /    /  ____\\               /    /  ____\\\ /                /    /  ____\\               /    /  ____\\\
|              |    |\/ 35 ||               |    |\/ 35 ||| |              |    |\/ 35 ||               |    |\/ 35 ||| |                |    |\/ 35 ||               |    |\/ 35 ||| |                |    |\/ 35 ||               |    |\/ 35 |||
|              |atan|------||               |atan|------||| |              |atan|------||               |atan|------||| |                |atan|------||               |atan|------||| |                |atan|------||               |atan|------|||
|    4 ____    |    \  5   /|     4 ____    |    \  5   /|| |    4 ____    |    \  5   /|     4 ____    |    \  5   /|| |      4 ____    |    \  5   /|     4 ____    |    \  5   /|| |      4 ____    |    \  5   /|     4 ____    |    \  5   /||
|x + \/ 15 *cos|------------| + I*\/ 15 *sin|------------||*|x + \/ 15 *cos|------------| - I*\/ 15 *sin|------------||*|x + - \/ 15 *cos|------------| + I*\/ 15 *sin|------------||*|x + - \/ 15 *cos|------------| - I*\/ 15 *sin|------------||
\              \     2      /               \     2      // \              \     2      /               \     2      // \                \     2      /               \     2      // \                \     2      /               \     2      //
(x+(154cos⁡(atan⁡(355)2)−154isin⁡(atan⁡(355)2)))(x+(154cos⁡(atan⁡(355)2)+154isin⁡(atan⁡(355)2)))(x+(−154cos⁡(atan⁡(355)2)+154isin⁡(atan⁡(355)2)))(x+(−154cos⁡(atan⁡(355)2)−154isin⁡(atan⁡(355)2)))\left(x + \left(\sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} - \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} + \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} + \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{15} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)} - \sqrt[4]{15} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{35}}{5} \right)}}{2} \right)}\right)\right) 
(((x + 15^(1/4)*cos(atan(sqrt(35)/5)/2) + i*15^(1/4)*sin(atan(sqrt(35)/5)/2))*(x + 15^(1/4)*cos(atan(sqrt(35)/5)/2) - i*15^(1/4)*sin(atan(sqrt(35)/5)/2)))*(x - 15^(1/4)*cos(atan(sqrt(35)/5)/2) + i*15^(1/4)*sin(atan(sqrt(35)/5)/2)))*(x - 15^(1/4)*cos(atan(sqrt(35)/5)/2) - i*15^(1/4)*sin(atan(sqrt(35)/5)/2))
Denominador común [src] 
      4      2
15 + y  - 5*y
y4−5y2+15y^{4} - 5 y^{2} + 15 
15 + y^4 - 5*y^2
Denominador racional [src] 
      4      2
15 + y  - 5*y
y4−5y2+15y^{4} - 5 y^{2} + 15 
15 + y^4 - 5*y^2
Unión de expresiones racionales [src] 
      2 /      2\
15 + y *\-5 + y /
y2(y2−5)+15y^{2} \left(y^{2} - 5\right) + 15 
15 + y^2*(-5 + y^2)
Combinatoria [src] 
      4      2
15 + y  - 5*y
y4−5y2+15y^{4} - 5 y^{2} + 15 
15 + y^4 - 5*y^2
Respuesta numérica [src] 
15.0 + y^4 - 5.0*y^2
15.0 + y^4 - 5.0*y^2
Compilar la expresión [src] 
      4      2
15 + y  - 5*y
y4−5y2+15y^{4} - 5 y^{2} + 15 
15 + y^4 - 5*y^2
Parte trigonométrica [src] 
      4      2
15 + y  - 5*y
y4−5y2+15y^{4} - 5 y^{2} + 15 
15 + y^4 - 5*y^2
Potencias [src] 
      4      2
15 + y  - 5*y
y4−5y2+15y^{4} - 5 y^{2} + 15 
15 + y^4 - 5*y^2
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