Sr Examen
Otras calculadoras
-
¿Cómo usar?
- La ecuación:
-
Ecuación x=(x+1)/2
- Ecuación (x-6)(-5x-9)=0
-
Ecuación x^2-14*x+33=0
-
Ecuación 5-x^2=0
- Expresar {x} en función de y en la ecuación:
- -17*x-12*y=-9
- 18*x+17*y=-14
- 14*x-10*y=-4
- -17*x-8*y=7
-
Expresiones idénticas
- kx^ dos +4x+4sqrt(tres)= cero
- kx al cuadrado más 4x más 4 raíz cuadrada de (3) es igual a 0
- kx en el grado dos más 4x más 4 raíz cuadrada de (tres) es igual a cero
- kx^2+4x+4√(3)=0
- kx2+4x+4sqrt(3)=0
- kx2+4x+4sqrt3=0
- kx²+4x+4sqrt(3)=0
- kx en el grado 2+4x+4sqrt(3)=0
- kx^2+4x+4sqrt3=0
- kx^2+4x+4sqrt(3)=O
-
Expresiones semejantes
- kx^2+4x-4sqrt(3)=0
- kx^2-4x+4sqrt(3)=0
-
Expresiones con funciones
- kx
- kx+9=6
- kx²-25=(3x+5)(3x-5)
kx^2+4x+4sqrt(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
$$a = k$$
$$b = 4$$
$$c = 4 \sqrt{3}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{- 16 \sqrt{3} k + 16} - 4}{2 k}$$
$$x_{2} = \frac{- \sqrt{- 16 \sqrt{3} k + 16} - 4}{2 k}$$
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
$$a = k$$
$$b = 4$$
$$c = 4 \sqrt{3}$$
, entonces
D = b^2 - 4 * a * c =
(4)^2 - 4 * (k) * (4*sqrt(3)) = 16 - 16*k*sqrt(3)
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{- 16 \sqrt{3} k + 16} - 4}{2 k}$$
$$x_{2} = \frac{- \sqrt{- 16 \sqrt{3} k + 16} - 4}{2 k}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(k x^{2} + 4 x\right) + 4 \sqrt{3} = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{k x^{2} + 4 x + 4 \sqrt{3}}{k} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{4}{k}$$
$$q = \frac{c}{a}$$
$$q = \frac{4 \sqrt{3}}{k}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{4}{k}$$
$$x_{1} x_{2} = \frac{4 \sqrt{3}}{k}$$
$$\left(k x^{2} + 4 x\right) + 4 \sqrt{3} = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{k x^{2} + 4 x + 4 \sqrt{3}}{k} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{4}{k}$$
$$q = \frac{c}{a}$$
$$q = \frac{4 \sqrt{3}}{k}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{4}{k}$$
$$x_{1} x_{2} = \frac{4 \sqrt{3}}{k}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$k x^{2} + 4 x + 4 \sqrt{3} = 0$$
Коэффициент при x равен
$$k$$
entonces son posibles los casos para k :
$$k < 0$$
$$k = 0$$
Consideremos todos los casos con detalles:
Con
$$k < 0$$
la ecuación será
$$- x^{2} + 4 x + 4 \sqrt{3} = 0$$
su solución
$$x = 2 - 2 \sqrt{1 + \sqrt{3}}$$
$$x = 2 + 2 \sqrt{1 + \sqrt{3}}$$
Con
$$k = 0$$
la ecuación será
$$4 x + 4 \sqrt{3} = 0$$
su solución
$$x = - \sqrt{3}$$
$$k x^{2} + 4 x + 4 \sqrt{3} = 0$$
Коэффициент при x равен
$$k$$
entonces son posibles los casos para k :
$$k < 0$$
$$k = 0$$
Consideremos todos los casos con detalles:
Con
$$k < 0$$
la ecuación será
$$- x^{2} + 4 x + 4 \sqrt{3} = 0$$
su solución
$$x = 2 - 2 \sqrt{1 + \sqrt{3}}$$
$$x = 2 + 2 \sqrt{1 + \sqrt{3}}$$
Con
$$k = 0$$
la ecuación será
$$4 x + 4 \sqrt{3} = 0$$
su solución
$$x = - \sqrt{3}$$
Gráfica
Respuesta rápida
[src]
/ / _______________________________ \ _______________________________ \ / _______________________________ \ _______________________________
| | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\
| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|
| 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *im(k)*sin|------------------------------------|
| \ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x1 = I*|- -------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ im (k) + re (k) im (k) + re (k) / im (k) + re (k) im (k) + re (k)
$$x_{1} = \frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + i \left(- \frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) + \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}$$
// _______________________________ \ _______________________________ \ / _______________________________ \ _______________________________
|| / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\
|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|
||2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| |2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *im(k)*sin|------------------------------------|
|\ \ 2 // \ 2 /| \ \ 2 // \ 2 /
x2 = I*|------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------
| 2 2 2 2 | 2 2 2 2
\ im (k) + re (k) im (k) + re (k) / im (k) + re (k) im (k) + re (k)
$$x_{2} = - \frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + i \left(\frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} - \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) - \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}$$
x2 = -(2*((-sqrt(3)*re(k) + 1)^2 + 3*im(k)^2)^(1/4)*cos(atan2(-sqrt(3)*im(k, -sqrt(3)*re(k) + 1)/2) + 2)*re(k)/(re(k)^2 + im(k)^2) + i*((2*((-sqrt(3)*re(k) + 1)^2 + 3*im(k)^2)^(1/4)*cos(atan2(-sqrt(3)*im(k), -sqrt(3)*re(k) + 1)/2) + 2)*im(k)/(re(k)^2 + im(k)^2) - 2*((-sqrt(3)*re(k) + 1)^2 + 3*im(k)^2)^(1/4)*sin(atan2(-sqrt(3)*im(k), -sqrt(3)*re(k) + 1)/2)*re(k)/(re(k)^2 + im(k)^2)) - 2*((-sqrt(3)*re(k) + 1)^2 + 3*im(k)^2)^(1/4)*sin(atan2(-sqrt(3)*im(k), -sqrt(3)*re(k) + 1)/2)*im(k)/(re(k)^2 + im(k)^2))
Suma y producto de raíces
[src]
suma
/ / _______________________________ \ _______________________________ \ / _______________________________ \ _______________________________ // _______________________________ \ _______________________________ \ / _______________________________ \ _______________________________ | | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\ || / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\ | | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/| || 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/| | 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *im(k)*sin|------------------------------------| ||2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| |2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *im(k)*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 (k) + re (k) im (k) + re (k) / im (k) + re (k) im (k) + re (k) \ im (k) + re (k) im (k) + re (k) / im (k) + re (k) im (k) + re (k)
$$\left(\frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + i \left(- \frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) + \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) + \left(- \frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + i \left(\frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} - \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) - \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right)$$
=
// _______________________________ \ _______________________________ \ / / _______________________________ \ _______________________________ \ / _______________________________ \ / _______________________________ \ || / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| || 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| ||2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| | 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| |2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) |\ \ 2 // \ 2 /| | \ \ 2 // \ 2 /| \ \ 2 // \ \ 2 // I*|------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------| + I*|- -------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------- | 2 2 2 2 | | 2 2 2 2 | 2 2 2 2 \ im (k) + re (k) im (k) + re (k) / \ im (k) + re (k) im (k) + re (k) / im (k) + re (k) im (k) + re (k)
$$\frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} - \frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + i \left(- \frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) + i \left(\frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} - \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right)$$
producto
/ / / _______________________________ \ _______________________________ \ / _______________________________ \ _______________________________ \ / // _______________________________ \ _______________________________ \ / _______________________________ \ _______________________________ \ | | | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | || / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | / 2 / / ___ ___ \\| / 2 / / ___ ___ \\| | | | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | || 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| 4 / / ___ \ 2 |atan2\-\/ 3 *im(k), 1 - \/ 3 *re(k)/|| | | 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| 2*|-1 + \/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *im(k)*sin|------------------------------------|| | ||2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*im(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *re(k)*sin|------------------------------------|| |2 + 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *cos|------------------------------------||*re(k) 2*\/ \1 - \/ 3 *re(k)/ + 3*im (k) *im(k)*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 (k) + re (k) im (k) + re (k) / im (k) + re (k) im (k) + re (k) / \ \ im (k) + re (k) im (k) + re (k) / im (k) + re (k) im (k) + re (k) /
$$\left(\frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + i \left(- \frac{2 \left(\sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} - 1\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) + \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) \left(- \frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} + i \left(\frac{\left(2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} + 2\right) \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}} - \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right) - \frac{2 \sqrt[4]{\left(- \sqrt{3} \operatorname{re}{\left(k\right)} + 1\right)^{2} + 3 \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \sqrt{3} \operatorname{im}{\left(k\right)},- \sqrt{3} \operatorname{re}{\left(k\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(k\right)}}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}\right)$$
=
___
4*\/ 3 *(-I*im(k) + re(k))
--------------------------
2 2
im (k) + re (k)
$$\frac{4 \sqrt{3} \left(\operatorname{re}{\left(k\right)} - i \operatorname{im}{\left(k\right)}\right)}{\left(\operatorname{re}{\left(k\right)}\right)^{2} + \left(\operatorname{im}{\left(k\right)}\right)^{2}}$$
4*sqrt(3)*(-i*im(k) + re(k))/(im(k)^2 + re(k)^2)