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- ax^2-7x-1=0
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ax^2-7x+1=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
$$b = -7$$
$$c = 1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{49 - 4 a} + 7}{2 a}$$
$$x_{2} = \frac{7 - \sqrt{49 - 4 a}}{2 a}$$
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
$$b = -7$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c =
(-7)^2 - 4 * (a) * (1) = 49 - 4*a
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{49 - 4 a} + 7}{2 a}$$
$$x_{2} = \frac{7 - \sqrt{49 - 4 a}}{2 a}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a x^{2} - 7 x + 1 = 0$$
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
$$- x^{2} - 7 x + 1 = 0$$
su solución
$$x = - \frac{7}{2} + \frac{\sqrt{53}}{2}$$
$$x = - \frac{\sqrt{53}}{2} - \frac{7}{2}$$
Con
$$a = 0$$
la ecuación será
$$1 - 7 x = 0$$
su solución
$$x = \frac{1}{7}$$
$$a x^{2} - 7 x + 1 = 0$$
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
$$- x^{2} - 7 x + 1 = 0$$
su solución
$$x = - \frac{7}{2} + \frac{\sqrt{53}}{2}$$
$$x = - \frac{\sqrt{53}}{2} - \frac{7}{2}$$
Con
$$a = 0$$
la ecuación será
$$1 - 7 x = 0$$
su solución
$$x = \frac{1}{7}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(a x^{2} - 7 x\right) + 1 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} - 7 x + 1}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{7}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{a}$$
$$x_{1} x_{2} = \frac{1}{a}$$
$$\left(a x^{2} - 7 x\right) + 1 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} - 7 x + 1}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{7}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{7}{a}$$
$$x_{1} x_{2} = \frac{1}{a}$$
Gráfica
Respuesta rápida
[src]
/ / _____________________________ \ _____________________________ \ / _____________________________ \ _____________________________
| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\
| |7 - \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| |7 - \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*re(a) \/ (49 - 4*re(a)) + 16*im (a) *im(a)*sin|-----------------------------|
| \ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x1 = I*|- ------------------------------------------------------------------------------- - -------------------------------------------------------------------------| + ------------------------------------------------------------------------------- - -------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{1} = \frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
/ / _____________________________ \ _____________________________ \ / _____________________________ \ _____________________________
| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\
| |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*re(a) \/ (49 - 4*re(a)) + 16*im (a) *im(a)*sin|-----------------------------|
| \ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x2 = I*|- ------------------------------------------------------------------------------- + -------------------------------------------------------------------------| + ------------------------------------------------------------------------------- + -------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{2} = \frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$
x2 = (((49 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*cos(atan2(-4*im(a, 49 - 4*re(a))/2) + 7)*re(a)/(2*(re(a)^2 + im(a)^2)) + i*(-(((49 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*cos(atan2(-4*im(a), 49 - 4*re(a))/2) + 7)*im(a)/(2*(re(a)^2 + im(a)^2)) + ((49 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*sin(atan2(-4*im(a), 49 - 4*re(a))/2)*re(a)/(2*(re(a)^2 + im(a)^2))) + ((49 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*sin(atan2(-4*im(a), 49 - 4*re(a))/2)*im(a)/(2*(re(a)^2 + im(a)^2)))
Suma y producto de raíces
[src]
suma
/ / _____________________________ \ _____________________________ \ / _____________________________ \ _____________________________ / / _____________________________ \ _____________________________ \ / _____________________________ \ _____________________________ | | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\ | | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\ | |7 - \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| |7 - \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*re(a) \/ (49 - 4*re(a)) + 16*im (a) *im(a)*sin|-----------------------------| | |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*re(a) \/ (49 - 4*re(a)) + 16*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 \ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$\left(\frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \left(\frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
=
/ / _____________________________ \ _____________________________ \ / / _____________________________ \ _____________________________ \ / _____________________________ \ / _____________________________ \ | | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| | |7 - \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*re(a) |7 - \/ (49 - 4*re(a)) + 16*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 \ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$\frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + i \left(- \frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
producto
/ / / _____________________________ \ _____________________________ \ / _____________________________ \ _____________________________ \ / / / _____________________________ \ _____________________________ \ / _____________________________ \ _____________________________ \ | | | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | | | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 49 - 4*re(a))\| | | |7 - \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| |7 - \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*re(a) \/ (49 - 4*re(a)) + 16*im (a) *im(a)*sin|-----------------------------|| | | |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*im(a) \/ (49 - 4*re(a)) + 16*im (a) *re(a)*sin|-----------------------------|| |7 + \/ (49 - 4*re(a)) + 16*im (a) *cos|-----------------------------||*re(a) \/ (49 - 4*re(a)) + 16*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 \ | \ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / \ \ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ /
$$\left(\frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) \left(\frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} + 7\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(49 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},49 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)} \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
=
-I*im(a) + re(a) ---------------- 2 2 im (a) + re (a)
$$\frac{\operatorname{re}{\left(a\right)} - i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
(-i*im(a) + re(a))/(im(a)^2 + re(a)^2)