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Ecuación x=(x+1)/2
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Ecuación x^2-14*x+33=0
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Ecuación x+18=8
- Expresar {x} en función de y en la ecuación:
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Expresiones idénticas
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- ( menos (x más uno) dividir por seis) en el grado dos más ((y menos cinco) dividir por cuatro) en el grado dos es igual a uno
- (-(x+1)/6)2+((y-5)/4)2=1
- -x+1/62+y-5/42=1
- (-(x+1)/6)²+((y-5)/4)²=1
- (-(x+1)/6) en el grado 2+((y-5)/4) en el grado 2=1
- -x+1/6^2+y-5/4^2=1
- (-(x+1) dividir por 6)^2+((y-5) dividir por 4)^2=1
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Expresiones semejantes
- (-(x+1)/6)^2-((y-5)/4)^2=1
- (-(x+1)/6)^2+((y+5)/4)^2=1
- (-(x-1)/6)^2+((y-5)/4)^2=1
- ((x+1)/6)^2+((y-5)/4)^2=1
(-(x+1)/6)^2+((y-5)/4)^2=1 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 /-x - 1\ /y - 5\ |------| + |-----| = 1 \ 6 / \ 4 /
$$\left(\frac{y - 5}{4}\right)^{2} + \left(\frac{- x - 1}{6}\right)^{2} = 1$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(\frac{y - 5}{4}\right)^{2} + \left(\frac{- x - 1}{6}\right)^{2} = 1$$
en
$$\left(\left(\frac{y - 5}{4}\right)^{2} + \left(\frac{- x - 1}{6}\right)^{2}\right) - 1 = 0$$
Abramos la expresión en la ecuación
$$\left(\left(\frac{y - 5}{4}\right)^{2} + \left(\frac{- x - 1}{6}\right)^{2}\right) - 1 = 0$$
Obtenemos la ecuación cuadrática
$$\frac{x^{2}}{36} + \frac{x}{18} + \frac{y^{2}}{16} - \frac{5 y}{8} + \frac{85}{144} = 0$$
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
$$a = \frac{1}{36}$$
$$b = \frac{1}{18}$$
$$c = \frac{y^{2}}{16} - \frac{5 y}{8} + \frac{85}{144}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = 18 \sqrt{- \frac{y^{2}}{144} + \frac{5 y}{72} - \frac{1}{16}} - 1$$
$$x_{2} = - 18 \sqrt{- \frac{y^{2}}{144} + \frac{5 y}{72} - \frac{1}{16}} - 1$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(\frac{y - 5}{4}\right)^{2} + \left(\frac{- x - 1}{6}\right)^{2} = 1$$
en
$$\left(\left(\frac{y - 5}{4}\right)^{2} + \left(\frac{- x - 1}{6}\right)^{2}\right) - 1 = 0$$
Abramos la expresión en la ecuación
$$\left(\left(\frac{y - 5}{4}\right)^{2} + \left(\frac{- x - 1}{6}\right)^{2}\right) - 1 = 0$$
Obtenemos la ecuación cuadrática
$$\frac{x^{2}}{36} + \frac{x}{18} + \frac{y^{2}}{16} - \frac{5 y}{8} + \frac{85}{144} = 0$$
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
$$a = \frac{1}{36}$$
$$b = \frac{1}{18}$$
$$c = \frac{y^{2}}{16} - \frac{5 y}{8} + \frac{85}{144}$$
, entonces
D = b^2 - 4 * a * c =
(1/18)^2 - 4 * (1/36) * (85/144 - 5*y/8 + y^2/16) = -1/16 - y^2/144 + 5*y/72
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = 18 \sqrt{- \frac{y^{2}}{144} + \frac{5 y}{72} - \frac{1}{16}} - 1$$
$$x_{2} = - 18 \sqrt{- \frac{y^{2}}{144} + \frac{5 y}{72} - \frac{1}{16}} - 1$$
Gráfica
Suma y producto de raíces
[src]
suma
__________________________________________________________________ __________________________________________________________________ __________________________________________________________________ __________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/| 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/| 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/| 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/|
3*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *cos|----------------------------------------------------------------| 3*I*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *sin|----------------------------------------------------------------| 3*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *cos|----------------------------------------------------------------| 3*I*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
-1 - ---------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------ + -1 + ---------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2
$$\left(- \frac{3 i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - \frac{3 \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - 1\right) + \left(\frac{3 i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} + \frac{3 \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - 1\right)$$
=
-2
$$-2$$
producto
/ __________________________________________________________________ __________________________________________________________________ \ / __________________________________________________________________ __________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/| 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/|| | 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/| 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/|| | 3*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *cos|----------------------------------------------------------------| 3*I*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *sin|----------------------------------------------------------------|| | 3*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *cos|----------------------------------------------------------------| 3*I*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *sin|----------------------------------------------------------------|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |-1 - ---------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------|*|-1 + ---------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 / \ 2 2 /
$$\left(- \frac{3 i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - \frac{3 \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - 1\right) \left(\frac{3 i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} + \frac{3 \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - 1\right)$$
=
2 2 85 45*re(y) 9*im (y) 9*re (y) 45*I*im(y) 9*I*im(y)*re(y) -- - -------- - -------- + -------- - ---------- + --------------- 4 2 4 4 2 2
$$\frac{9 \left(\operatorname{re}{\left(y\right)}\right)^{2}}{4} + \frac{9 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{2} - \frac{45 \operatorname{re}{\left(y\right)}}{2} - \frac{9 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{4} - \frac{45 i \operatorname{im}{\left(y\right)}}{2} + \frac{85}{4}$$
85/4 - 45*re(y)/2 - 9*im(y)^2/4 + 9*re(y)^2/4 - 45*i*im(y)/2 + 9*i*im(y)*re(y)/2
Respuesta rápida
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/| 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/|
3*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *cos|----------------------------------------------------------------| 3*I*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 /
x1 = -1 - ---------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{1} = - \frac{3 i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - \frac{3 \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - 1$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/| 4 / 2 / 2 2 \ |atan2\10*im(y) - 2*im(y)*re(y), -9 + im (y) - re (y) + 10*re(y)/|
3*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *cos|----------------------------------------------------------------| 3*I*\/ (10*im(y) - 2*im(y)*re(y)) + \-9 + im (y) - re (y) + 10*re(y)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 /
x2 = -1 + ---------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{2} = \frac{3 i \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} + \frac{3 \sqrt[4]{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 10 \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + 10 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} - 9 \right)}}{2} \right)}}{2} - 1$$
x2 = 3*i*((-2*re(y)*im(y) + 10*im(y))^2 + (-re(y)^2 + 10*re(y) + im(y)^2 - 9)^2)^(1/4)*sin(atan2(-2*re(y)*im(y) + 10*im(y, -re(y)^2 + 10*re(y) + im(y)^2 - 9)/2)/2 + 3*((-2*re(y)*im(y) + 10*im(y))^2 + (-re(y)^2 + 10*re(y) + im(y)^2 - 9)^2)^(1/4)*cos(atan2(-2*re(y)*im(y) + 10*im(y), -re(y)^2 + 10*re(y) + im(y)^2 - 9)/2)/2 - 1)