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Expresiones semejantes
- (2*x+3*y*i)*(2*x+3*y*i)+x*i=97+2*i
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- (2*x-3*y*i)*(2*x+3*y*i)-x*i=97+2*i
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(2*x-3*y*i)*(2*x+3*y*i)+x*i=97+2*i la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
(2*x - 3*y*I)*(2*x + 3*y*I) + x*I = 97 + 2*I
$$i x + \left(2 x - i 3 y\right) \left(2 x + i 3 y\right) = 97 + 2 i$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$i x + \left(2 x - i 3 y\right) \left(2 x + i 3 y\right) = 97 + 2 i$$
en
$$\left(i x + \left(2 x - i 3 y\right) \left(2 x + i 3 y\right)\right) + \left(-97 - 2 i\right) = 0$$
Abramos la expresión en la ecuación
$$\left(i x + \left(2 x - i 3 y\right) \left(2 x + i 3 y\right)\right) + \left(-97 - 2 i\right) = 0$$
Obtenemos la ecuación cuadrática
$$4 x^{2} + i x + 9 y^{2} - 97 - 2 i = 0$$
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
$$a = 4$$
$$b = i$$
$$c = 9 y^{2} - 97 - 2 i$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{- 144 y^{2} + 1551 + 32 i}}{8} - \frac{i}{8}$$
$$x_{2} = - \frac{\sqrt{- 144 y^{2} + 1551 + 32 i}}{8} - \frac{i}{8}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$i x + \left(2 x - i 3 y\right) \left(2 x + i 3 y\right) = 97 + 2 i$$
en
$$\left(i x + \left(2 x - i 3 y\right) \left(2 x + i 3 y\right)\right) + \left(-97 - 2 i\right) = 0$$
Abramos la expresión en la ecuación
$$\left(i x + \left(2 x - i 3 y\right) \left(2 x + i 3 y\right)\right) + \left(-97 - 2 i\right) = 0$$
Obtenemos la ecuación cuadrática
$$4 x^{2} + i x + 9 y^{2} - 97 - 2 i = 0$$
Es la ecuación de la forma
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
$$a = 4$$
$$b = i$$
$$c = 9 y^{2} - 97 - 2 i$$
, entonces
D = b^2 - 4 * a * c =
(i)^2 - 4 * (4) * (-97 - 2*i + 9*y^2) = 1551 - 144*y^2 + 32*i
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{- 144 y^{2} + 1551 + 32 i}}{8} - \frac{i}{8}$$
$$x_{2} = - \frac{\sqrt{- 144 y^{2} + 1551 + 32 i}}{8} - \frac{i}{8}$$
Gráfica
Respuesta rápida
[src]
/ _____________________________________________________________ \ _____________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|
| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *cos|-----------------------------------------------------------|
| 1 \ 2 /| \ 2 /
x1 = I*|- - - ----------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------
\ 8 8 / 8
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right) - \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8}$$
/ _____________________________________________________________ \ _____________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|
| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *cos|-----------------------------------------------------------|
| 1 \ 2 /| \ 2 /
x2 = I*|- - + ----------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------
\ 8 8 / 8
$$x_{2} = i \left(\frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right) + \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8}$$
x2 = i*(((-288*re(y)*im(y) + 32)^2 + (-144*re(y)^2 + 144*im(y)^2 + 1551)^2)^(1/4)*sin(atan2(-288*re(y)*im(y + 32, -144*re(y)^2 + 144*im(y)^2 + 1551)/2)/8 - 1/8) + ((-288*re(y)*im(y) + 32)^2 + (-144*re(y)^2 + 144*im(y)^2 + 1551)^2)^(1/4)*cos(atan2(-288*re(y)*im(y) + 32, -144*re(y)^2 + 144*im(y)^2 + 1551)/2)/8)
Suma y producto de raíces
[src]
suma
/ _____________________________________________________________ \ _____________________________________________________________ / _____________________________________________________________ \ _____________________________________________________________ | / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\ | 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/| | 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/| | \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *cos|-----------------------------------------------------------| | \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *cos|-----------------------------------------------------------| | 1 \ 2 /| \ 2 / | 1 \ 2 /| \ 2 / I*|- - - ----------------------------------------------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------------------------------------------------- + I*|- - + ----------------------------------------------------------------------------------------------------------------------------------| + ---------------------------------------------------------------------------------------------------------------------------------- \ 8 8 / 8 \ 8 8 / 8
$$\left(i \left(- \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right) - \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8}\right) + \left(i \left(\frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right) + \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8}\right)$$
=
/ _____________________________________________________________ \ / _____________________________________________________________ \ | / 2 / / 2 2 \\| | / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| | 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| | \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| | \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| | 1 \ 2 /| | 1 \ 2 /| I*|- - - ----------------------------------------------------------------------------------------------------------------------------------| + I*|- - + ----------------------------------------------------------------------------------------------------------------------------------| \ 8 8 / \ 8 8 /
$$i \left(- \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right) + i \left(\frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right)$$
producto
/ / _____________________________________________________________ \ _____________________________________________________________ \ / / _____________________________________________________________ \ _____________________________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| | | 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| 4 / 2 / 2 2 \ |atan2\32 - 288*im(y)*re(y), 1551 - 144*re (y) + 144*im (y)/|| | | \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *cos|-----------------------------------------------------------|| | | \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *sin|-----------------------------------------------------------|| \/ (32 - 288*im(y)*re(y)) + \1551 - 144*re (y) + 144*im (y)/ *cos|-----------------------------------------------------------|| | | 1 \ 2 /| \ 2 /| | | 1 \ 2 /| \ 2 /| |I*|- - - ----------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------|*|I*|- - + ----------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------| \ \ 8 8 / 8 / \ \ 8 8 / 8 /
$$\left(i \left(- \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right) - \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8}\right) \left(i \left(\frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8} - \frac{1}{8}\right) + \frac{\sqrt[4]{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32\right)^{2} + \left(- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 288 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 32,- 144 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 144 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1551 \right)}}{2} \right)}}{8}\right)$$
=
2 2 97 9*im (y) I 9*re (y) 9*I*im(y)*re(y) - -- - -------- - - + -------- + --------------- 4 4 2 4 2
$$\frac{9 \left(\operatorname{re}{\left(y\right)}\right)^{2}}{4} + \frac{9 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{2} - \frac{9 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{4} - \frac{97}{4} - \frac{i}{2}$$
-97/4 - 9*im(y)^2/4 - i/2 + 9*re(y)^2/4 + 9*i*im(y)*re(y)/2