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Expresiones semejantes
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(x^(2))+sqrt(a^2+4a)*x+(a+1)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
__________ 2 / 2 x + \/ a + 4*a *x + a + 1 = 0
$$\left(a + 1\right) + \left(x^{2} + x \sqrt{a^{2} + 4 a}\right) = 0$$
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
$$a = 1$$
$$b = \sqrt{a^{2} + 4 a}$$
$$c = a + 1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{a^{2} - 4}}{2} - \frac{\sqrt{a^{2} + 4 a}}{2}$$
$$x_{2} = - \frac{\sqrt{a^{2} - 4}}{2} - \frac{\sqrt{a^{2} + 4 a}}{2}$$
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
$$a = 1$$
$$b = \sqrt{a^{2} + 4 a}$$
$$c = a + 1$$
, entonces
D = b^2 - 4 * a * c =
(sqrt(a^2 + 4*a))^2 - 4 * (1) * (1 + a) = -4 + a^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{a^{2} - 4}}{2} - \frac{\sqrt{a^{2} + 4 a}}{2}$$
$$x_{2} = - \frac{\sqrt{a^{2} - 4}}{2} - \frac{\sqrt{a^{2} + 4 a}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \sqrt{a^{2} + 4 a}$$
$$q = \frac{c}{a}$$
$$q = a + 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \sqrt{a^{2} + 4 a}$$
$$x_{1} x_{2} = a + 1$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \sqrt{a^{2} + 4 a}$$
$$q = \frac{c}{a}$$
$$q = a + 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \sqrt{a^{2} + 4 a}$$
$$x_{1} x_{2} = a + 1$$
Gráfica
Respuesta rápida
[src]
/ ______________________________________________________________________ ___________________________________________ \ ______________________________________________________________________ ___________________________________________
| / 2 / / 2 \\ / 2 / / 2 2 \\| / 2 / / 2 \\ / 2 / / 2 2 \\
| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/|| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/|
| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *cos|--------------------------------------------------------------------| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------|
| \ 2 / \ 2 /| \ 2 / \ 2 /
x1 = I*|- ---------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}$$
/ ___________________________________________ ______________________________________________________________________ \ ___________________________________________ ______________________________________________________________________
| / 2 / / 2 2 \\ / 2 / / 2 \\| / 2 / / 2 2 \\ / 2 / / 2 \\
|4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/|
|\/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------|| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *cos|--------------------------------------------------------------------|
| \ 2 / \ 2 /| \ 2 / \ 2 /
x2 = I*|----------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$x_{2} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}$$
x2 = i*(-(((re(a) + 4)*re(a) - im(a)^2)^2 + ((re(a) + 4)*im(a) + re(a)*im(a))^2)^(1/4)*sin(atan2((re(a) + 4)*im(a) + re(a)*im(a, (re(a) + 4)*re(a) - im(a)^2)/2)/2 + ((re(a)^2 - im(a)^2 - 4)^2 + 4*re(a)^2*im(a)^2)^(1/4)*sin(atan2(2*re(a)*im(a), re(a)^2 - im(a)^2 - 4)/2)/2) - (((re(a) + 4)*re(a) - im(a)^2)^2 + ((re(a) + 4)*im(a) + re(a)*im(a))^2)^(1/4)*cos(atan2((re(a) + 4)*im(a) + re(a)*im(a), (re(a) + 4)*re(a) - im(a)^2)/2)/2 + ((re(a)^2 - im(a)^2 - 4)^2 + 4*re(a)^2*im(a)^2)^(1/4)*cos(atan2(2*re(a)*im(a), re(a)^2 - im(a)^2 - 4)/2)/2)
Suma y producto de raíces
[src]
suma
/ ______________________________________________________________________ ___________________________________________ \ ______________________________________________________________________ ___________________________________________ / ___________________________________________ ______________________________________________________________________ \ ___________________________________________ ______________________________________________________________________ | / 2 / / 2 \\ / 2 / / 2 2 \\| / 2 / / 2 \\ / 2 / / 2 2 \\ | / 2 / / 2 2 \\ / 2 / / 2 \\| / 2 / / 2 2 \\ / 2 / / 2 \\ | 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/|| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| |4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| | \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *cos|--------------------------------------------------------------------| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------| |\/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------|| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *cos|--------------------------------------------------------------------| | \ 2 / \ 2 /| \ 2 / \ 2 / | \ 2 / \ 2 /| \ 2 / \ 2 / I*|- ---------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------- + I*|----------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------- \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) + \left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right)$$
=
/ ___________________________________________ ______________________________________________________________________ \ / ______________________________________________________________________ ___________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 \\| | / 2 / / 2 \\ / 2 / / 2 2 \\| |4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/|| | 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/|| ______________________________________________________________________ |\/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------|| | \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| / 2 / / 2 \\ | \ 2 / \ 2 /| | \ 2 / \ 2 /| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| I*|----------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------| + I*|- ---------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------| - \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *cos|--------------------------------------------------------------------| \ 2 2 / \ 2 2 / \ 2 /
$$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) + i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) - \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}$$
producto
/ / ______________________________________________________________________ ___________________________________________ \ ______________________________________________________________________ ___________________________________________ \ / / ___________________________________________ ______________________________________________________________________ \ ___________________________________________ ______________________________________________________________________ \ | | / 2 / / 2 \\ / 2 / / 2 2 \\| / 2 / / 2 \\ / 2 / / 2 2 \\| | | / 2 / / 2 2 \\ / 2 / / 2 \\| / 2 / / 2 2 \\ / 2 / / 2 \\| | | 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/|| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/|| | |4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), -4 + re (a) - im (a)/| 4 / / 2 \ 2 |atan2\(4 + re(a))*im(a) + im(a)*re(a), - im (a) + (4 + re(a))*re(a)/|| | | \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *cos|--------------------------------------------------------------------| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------|| | |\/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *sin|--------------------------------------------------------------------|| \/ \-4 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------| \/ \- im (a) + (4 + re(a))*re(a)/ + ((4 + re(a))*im(a) + im(a)*re(a)) *cos|--------------------------------------------------------------------|| | | \ 2 / \ 2 /| \ 2 / \ 2 /| | | \ 2 / \ 2 /| \ 2 / \ 2 /| |I*|- ---------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------|*|I*|----------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------| \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) \left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 \right)}}{2} \right)}}{2}\right)$$
=
1 + I*im(a) + re(a)
$$\operatorname{re}{\left(a\right)} + i \operatorname{im}{\left(a\right)} + 1$$
1 + i*im(a) + re(a)