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x^4-3x^2+4=0

x^4-3x^2+4=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 4      2        
x  - 3*x  + 4 = 0
(x4−3x2)+4=0\left(x^{4} - 3 x^{2}\right) + 4 = 0
Simplificar
Solución detallada
Tenemos la ecuación:
(x4−3x2)+4=0\left(x^{4} - 3 x^{2}\right) + 4 = 0
Sustituimos
v=x2v = x^{2}
entonces la ecuación será así:
v2−3v+4=0v^{2} - 3 v + 4 = 0
Es la ecuación de la forma
a*v^2 + b*v + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
v1=D−b2av_{1} = \frac{\sqrt{D} - b}{2 a}
v2=−D−b2av_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=−3b = -3
c=4c = 4
, entonces
D = b^2 - 4 * a * c = 

(-3)^2 - 4 * (1) * (4) = -7

Como D < 0 la ecuación
no tiene raíces reales,
pero hay raíces complejas.
v1 = (-b + sqrt(D)) / (2*a)

v2 = (-b - sqrt(D)) / (2*a)

o
v1=32+7i2v_{1} = \frac{3}{2} + \frac{\sqrt{7} i}{2}
v2=32−7i2v_{2} = \frac{3}{2} - \frac{\sqrt{7} i}{2}
Entonces la respuesta definitiva es:
Como
v=x2v = x^{2}
entonces
x1=v1x_{1} = \sqrt{v_{1}}
x2=−v1x_{2} = - \sqrt{v_{1}}
x3=v2x_{3} = \sqrt{v_{2}}
x4=−v2x_{4} = - \sqrt{v_{2}}
entonces:
x1=x_{1} =
01+(32+7i2)121=32+7i2\frac{0}{1} + \frac{\left(\frac{3}{2} + \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{3}{2} + \frac{\sqrt{7} i}{2}}
x2=x_{2} =
01+(−1)(32+7i2)121=−32+7i2\frac{0}{1} + \frac{\left(-1\right) \left(\frac{3}{2} + \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{3}{2} + \frac{\sqrt{7} i}{2}}
x3=x_{3} =
01+(32−7i2)121=32−7i2\frac{0}{1} + \frac{\left(\frac{3}{2} - \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{3}{2} - \frac{\sqrt{7} i}{2}}
x4=x_{4} =
01+(−1)(32−7i2)121=−32−7i2\frac{0}{1} + \frac{\left(-1\right) \left(\frac{3}{2} - \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{3}{2} - \frac{\sqrt{7} i}{2}}
Gráfica
-3.5-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.5020
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
                /    /  ___\\              /    /  ___\\
                |    |\/ 7 ||              |    |\/ 7 ||
                |atan|-----||              |atan|-----||
         ___    |    \  3  /|       ___    |    \  3  /|
x1 = - \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------|
                \     2     /              \     2     /
x1=−2cos⁡(atan⁡(73)2)−2isin⁡(atan⁡(73)2)x_{1} = - \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} 
                /    /  ___\\              /    /  ___\\
                |    |\/ 7 ||              |    |\/ 7 ||
                |atan|-----||              |atan|-----||
         ___    |    \  3  /|       ___    |    \  3  /|
x2 = - \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------|
                \     2     /              \     2     /
x2=−2cos⁡(atan⁡(73)2)+2isin⁡(atan⁡(73)2)x_{2} = - \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} 
              /    /  ___\\              /    /  ___\\
              |    |\/ 7 ||              |    |\/ 7 ||
              |atan|-----||              |atan|-----||
       ___    |    \  3  /|       ___    |    \  3  /|
x3 = \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------|
              \     2     /              \     2     /
x3=2cos⁡(atan⁡(73)2)−2isin⁡(atan⁡(73)2)x_{3} = \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} 
              /    /  ___\\              /    /  ___\\
              |    |\/ 7 ||              |    |\/ 7 ||
              |atan|-----||              |atan|-----||
       ___    |    \  3  /|       ___    |    \  3  /|
x4 = \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------|
              \     2     /              \     2     /
x4=2cos⁡(atan⁡(73)2)+2isin⁡(atan⁡(73)2)x_{4} = \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} 
x4 = sqrt(2)*cos(atan(sqrt(7)/3)/2) + sqrt(2)*i*sin(atan(sqrt(7)/3)/2)
Suma y producto de raíces [src] 
suma
           /    /  ___\\              /    /  ___\\              /    /  ___\\              /    /  ___\\            /    /  ___\\              /    /  ___\\            /    /  ___\\              /    /  ___\\
           |    |\/ 7 ||              |    |\/ 7 ||              |    |\/ 7 ||              |    |\/ 7 ||            |    |\/ 7 ||              |    |\/ 7 ||            |    |\/ 7 ||              |    |\/ 7 ||
           |atan|-----||              |atan|-----||              |atan|-----||              |atan|-----||            |atan|-----||              |atan|-----||            |atan|-----||              |atan|-----||
    ___    |    \  3  /|       ___    |    \  3  /|       ___    |    \  3  /|       ___    |    \  3  /|     ___    |    \  3  /|       ___    |    \  3  /|     ___    |    \  3  /|       ___    |    \  3  /|
- \/ 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     /
((2cos⁡(atan⁡(73)2)−2isin⁡(atan⁡(73)2))+((−2cos⁡(atan⁡(73)2)−2isin⁡(atan⁡(73)2))+(−2cos⁡(atan⁡(73)2)+2isin⁡(atan⁡(73)2))))+(2cos⁡(atan⁡(73)2)+2isin⁡(atan⁡(73)2))\left(\left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) + \left(\left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right)\right) + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) 
=
0
00 
producto
/           /    /  ___\\              /    /  ___\\\ /           /    /  ___\\              /    /  ___\\\ /         /    /  ___\\              /    /  ___\\\ /         /    /  ___\\              /    /  ___\\\
|           |    |\/ 7 ||              |    |\/ 7 ||| |           |    |\/ 7 ||              |    |\/ 7 ||| |         |    |\/ 7 ||              |    |\/ 7 ||| |         |    |\/ 7 ||              |    |\/ 7 |||
|           |atan|-----||              |atan|-----||| |           |atan|-----||              |atan|-----||| |         |atan|-----||              |atan|-----||| |         |atan|-----||              |atan|-----|||
|    ___    |    \  3  /|       ___    |    \  3  /|| |    ___    |    \  3  /|       ___    |    \  3  /|| |  ___    |    \  3  /|       ___    |    \  3  /|| |  ___    |    \  3  /|       ___    |    \  3  /||
|- \/ 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     //
(−2cos⁡(atan⁡(73)2)−2isin⁡(atan⁡(73)2))(−2cos⁡(atan⁡(73)2)+2isin⁡(atan⁡(73)2))(2cos⁡(atan⁡(73)2)−2isin⁡(atan⁡(73)2))(2cos⁡(atan⁡(73)2)+2isin⁡(atan⁡(73)2))\left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) 
=
                       /  ___\
                       |\/ 7 |
  /        ___\  I*atan|-----|
  |3   I*\/ 7 |        \  3  /
4*|- - -------|*e             
  \4      4   /
4(34−7i4)eiatan⁡(73)4 \left(\frac{3}{4} - \frac{\sqrt{7} i}{4}\right) e^{i \operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}} 
4*(3/4 - i*sqrt(7)/4)*exp(i*atan(sqrt(7)/3))
Respuesta numérica [src] 
x1 = 1.3228756555323 + 0.5*i
x2 = -1.3228756555323 + 0.5*i
x3 = -1.3228756555323 - 0.5*i
x4 = 1.3228756555323 - 0.5*i
x4 = 1.3228756555323 - 0.5*i
Gráfico
x^4-3x^2+4=0 la ecuación
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