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- La ecuación:
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Ecuación x^4+2*x^2-8=0
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Ecuación x*(x^2+8*x+16)=5*(x+4)
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Ecuación -25-x=-17
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Ecuación x^2+3x=4
- Expresar {x} en función de y en la ecuación:
- -18*x-2*y=9
- -14*x+10*y=19
- -9*x+1*y=3
- -17*x+1*y=-11
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Expresiones idénticas
- ax^ dos +2x+ tres = cero
- ax al cuadrado más 2x más 3 es igual a 0
- ax en el grado dos más 2x más tres es igual a cero
- ax2+2x+3=0
- ax²+2x+3=0
- ax en el grado 2+2x+3=0
- ax^2+2x+3=O
-
Expresiones semejantes
- ax^2-2x+3=0
- ax^2+2x-3=0
-
Expresiones con funciones
- ax
- ax^2+bx+c=0
- ax+b=cx+d
- ax^2+8*x-4*a-16=0
- ax^2+ax+3=0
- ax+8y=20
ax^2+2x+3=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$b = 2$$
$$c = 3$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{4 - 12 a} - 2}{2 a}$$
$$x_{2} = \frac{- \sqrt{4 - 12 a} - 2}{2 a}$$
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
True
$$b = 2$$
$$c = 3$$
, entonces
D = b^2 - 4 * a * c =
(2)^2 - 4 * (a) * (3) = 4 - 12*a
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{4 - 12 a} - 2}{2 a}$$
$$x_{2} = \frac{- \sqrt{4 - 12 a} - 2}{2 a}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a x^{2} + 2 x + 3 = 0$$
Коэффициент при x равен
$$a$$
entonces son posibles los casos para a :
$$a < 0$$
$$a = 0$$
Consideremos todos los casos con detalles:
Con
$$a < 0$$
la ecuación será
$$- x^{2} + 2 x + 3 = 0$$
su solución
$$x = -1$$
$$x = 3$$
Con
$$a = 0$$
la ecuación será
$$2 x + 3 = 0$$
su solución
$$x = - \frac{3}{2}$$
$$a x^{2} + 2 x + 3 = 0$$
Коэффициент при x равен
$$a$$
entonces son posibles los casos para a :
$$a < 0$$
$$a = 0$$
Consideremos todos los casos con detalles:
Con
$$a < 0$$
la ecuación será
$$- x^{2} + 2 x + 3 = 0$$
su solución
$$x = -1$$
$$x = 3$$
Con
$$a = 0$$
la ecuación será
$$2 x + 3 = 0$$
su solución
$$x = - \frac{3}{2}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(a x^{2} + 2 x\right) + 3 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + 2 x + 3}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2}{a}$$
$$x_{1} x_{2} = \frac{3}{a}$$
$$\left(a x^{2} + 2 x\right) + 3 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + 2 x + 3}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2}{a}$$
$$x_{1} x_{2} = \frac{3}{a}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ / ___________________________ \ ___________________________ \ / ___________________________ \ ___________________________ // ___________________________ \ ___________________________ \ / ___________________________ \ ___________________________ | | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\ || 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\ | |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) \/ (1 - 3*re(a)) + 9*im (a) *im(a)*sin|----------------------------| ||1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| |1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) \/ (1 - 3*re(a)) + 9*im (a) *im(a)*sin|----------------------------| | \ \ 2 // \ 2 /| \ \ 2 // \ 2 / |\ \ 2 // \ 2 /| \ \ 2 // \ 2 / I*|- ----------------------------------------------------------------------------- + ----------------------------------------------------------------------| + ----------------------------------------------------------------------------- + ---------------------------------------------------------------------- + I*|---------------------------------------------------------------------------- - ----------------------------------------------------------------------| - ---------------------------------------------------------------------------- - ---------------------------------------------------------------------- | 2 2 2 2 | 2 2 2 2 | 2 2 2 2 | 2 2 2 2 \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a) \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$\left(\frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \left(- \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
=
// ___________________________ \ ___________________________ \ / / ___________________________ \ ___________________________ \ / ___________________________ \ / ___________________________ \ || 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| ||1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| | |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) |1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) |\ \ 2 // \ 2 /| | \ \ 2 // \ 2 /| \ \ 2 // \ \ 2 // I*|---------------------------------------------------------------------------- - ----------------------------------------------------------------------| + I*|- ----------------------------------------------------------------------------- + ----------------------------------------------------------------------| + ----------------------------------------------------------------------------- - ---------------------------------------------------------------------------- | 2 2 2 2 | | 2 2 2 2 | 2 2 2 2 \ im (a) + re (a) im (a) + re (a) / \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$\frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + i \left(\frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
producto
/ / / ___________________________ \ ___________________________ \ / ___________________________ \ ___________________________ \ / // ___________________________ \ ___________________________ \ / ___________________________ \ ___________________________ \ | | | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | || 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | | |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) \/ (1 - 3*re(a)) + 9*im (a) *im(a)*sin|----------------------------|| | ||1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| |1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) \/ (1 - 3*re(a)) + 9*im (a) *im(a)*sin|----------------------------|| | | \ \ 2 // \ 2 /| \ \ 2 // \ 2 /| | |\ \ 2 // \ 2 /| \ \ 2 // \ 2 /| |I*|- ----------------------------------------------------------------------------- + ----------------------------------------------------------------------| + ----------------------------------------------------------------------------- + ----------------------------------------------------------------------|*|I*|---------------------------------------------------------------------------- - ----------------------------------------------------------------------| - ---------------------------------------------------------------------------- - ----------------------------------------------------------------------| | | 2 2 2 2 | 2 2 2 2 | | | 2 2 2 2 | 2 2 2 2 | \ \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a) / \ \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a) /
$$\left(\frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) \left(- \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
=
3*re(a) - 3*I*im(a)
-------------------
2 2
im (a) + re (a)
$$\frac{3 \operatorname{re}{\left(a\right)} - 3 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
(3*re(a) - 3*i*im(a))/(im(a)^2 + re(a)^2)
Respuesta rápida
[src]
/ / ___________________________ \ ___________________________ \ / ___________________________ \ ___________________________
| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\
| |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| |-1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) \/ (1 - 3*re(a)) + 9*im (a) *im(a)*sin|----------------------------|
| \ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x1 = I*|- ----------------------------------------------------------------------------- + ----------------------------------------------------------------------| + ----------------------------------------------------------------------------- + ----------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$x_{1} = \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(- \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
// ___________________________ \ ___________________________ \ / ___________________________ \ ___________________________
|| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| | 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\| 4 / 2 2 /atan2(-3*im(a), 1 - 3*re(a))\
||1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*im(a) \/ (1 - 3*re(a)) + 9*im (a) *re(a)*sin|----------------------------|| |1 + \/ (1 - 3*re(a)) + 9*im (a) *cos|----------------------------||*re(a) \/ (1 - 3*re(a)) + 9*im (a) *im(a)*sin|----------------------------|
|\ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x2 = I*|---------------------------------------------------------------------------- - ----------------------------------------------------------------------| - ---------------------------------------------------------------------------- - ----------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$x_{2} = - \frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) - \frac{\sqrt[4]{\left(1 - 3 \operatorname{re}{\left(a\right)}\right)^{2} + 9 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \operatorname{im}{\left(a\right)},1 - 3 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
x2 = -(((1 - 3*re(a))^2 + 9*im(a)^2)^(1/4)*cos(atan2(-3*im(a, 1 - 3*re(a))/2) + 1)*re(a)/(re(a)^2 + im(a)^2) + i*((((1 - 3*re(a))^2 + 9*im(a)^2)^(1/4)*cos(atan2(-3*im(a), 1 - 3*re(a))/2) + 1)*im(a)/(re(a)^2 + im(a)^2) - ((1 - 3*re(a))^2 + 9*im(a)^2)^(1/4)*sin(atan2(-3*im(a), 1 - 3*re(a))/2)*re(a)/(re(a)^2 + im(a)^2)) - ((1 - 3*re(a))^2 + 9*im(a)^2)^(1/4)*sin(atan2(-3*im(a), 1 - 3*re(a))/2)*im(a)/(re(a)^2 + im(a)^2))