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La ecuación:
Ecuación 56/x=(10+6)/2
Ecuación 15-4(7-x)=11
Ecuación 3^8-x=27
Ecuación 6,4(y-12,8)=3,2
Expresar {x} en función de y en la ecuación:
20*x+7*y=-9
-13*x-10*y=5
15*x+7*y=2
20*x+9*y=17
Expresiones idénticas
(-(x+ uno)/ seis)^ dos +((y- cinco)/ cuatro)^ dos = uno
( menos (x más 1) dividir por 6) al cuadrado más ((y menos 5) dividir por 4) al cuadrado es igual a 1
( 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
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

Simplificar

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

\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)

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 [src]

                __________________________________________________________________                                                                                 __________________________________________________________________                                                                      
               /                                                                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

                __________________________________________________________________                                                                                 __________________________________________________________________                                                                      
               /                                                                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)

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Expresiones idénticas

Expresiones semejantes

(-(x+1)/6)^2+((y-5)/4)^2=1 la ecuación

Solución numérica:

Solución