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x^2-2*x+2*y^2+8*y+3=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 2            2              
x  - 2*x + 2*y  + 8*y + 3 = 0
(8y+(2y2+(x2−2x)))+3=0\left(8 y + \left(2 y^{2} + \left(x^{2} - 2 x\right)\right)\right) + 3 = 0
Simplificar
Solución detallada
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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=−2b = -2
c=2y2+8y+3c = 2 y^{2} + 8 y + 3
, entonces
D = b^2 - 4 * a * c = 

(-2)^2 - 4 * (1) * (3 + 2*y^2 + 8*y) = -8 - 32*y - 8*y^2

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
x1=−8y2−32y−82+1x_{1} = \frac{\sqrt{- 8 y^{2} - 32 y - 8}}{2} + 1
x2=1−−8y2−32y−82x_{2} = 1 - \frac{\sqrt{- 8 y^{2} - 32 y - 8}}{2}
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=−2p = -2
q=caq = \frac{c}{a}
q=2y2+8y+3q = 2 y^{2} + 8 y + 3
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=2x_{1} + x_{2} = 2
x1x2=2y2+8y+3x_{1} x_{2} = 2 y^{2} + 8 y + 3
Gráfica
Respuesta rápida [src] 
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            /                                                                   2     /     /                                             2          2   \\        /                                                                   2     /     /                                             2          2   \\
         4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|     4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|
x1 = 1 - \/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *cos|-------------------------------------------------------------------| - I*\/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *sin|-------------------------------------------------------------------|
                                                                                      \                                 2                                 /                                                                                  \                                 2                                 /
x1=−i(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24sin⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)−(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24cos⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+1x_{1} = - i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} - \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + 1 
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            /                                                                   2     /     /                                             2          2   \\        /                                                                   2     /     /                                             2          2   \\
         4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|     4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|
x2 = 1 + \/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *cos|-------------------------------------------------------------------| + I*\/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *sin|-------------------------------------------------------------------|
                                                                                      \                                 2                                 /                                                                                  \                                 2                                 /
x2=i(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24sin⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24cos⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+1x_{2} = i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + 1 
x2 = i*((-4*re(y)*im(y) - 8*im(y))^2 + (-2*re(y)^2 - 8*re(y) + 2*im(y)^2 - 2)^2)^(1/4)*sin(atan2(-4*re(y)*im(y) - 8*im(y, -2*re(y)^2 - 8*re(y) + 2*im(y)^2 - 2)/2) + ((-4*re(y)*im(y) - 8*im(y))^2 + (-2*re(y)^2 - 8*re(y) + 2*im(y)^2 - 2)^2)^(1/4)*cos(atan2(-4*re(y)*im(y) - 8*im(y), -2*re(y)^2 - 8*re(y) + 2*im(y)^2 - 2)/2) + 1)
Suma y producto de raíces [src] 
suma
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       /                                                                   2     /     /                                             2          2   \\        /                                                                   2     /     /                                             2          2   \\          /                                                                   2     /     /                                             2          2   \\        /                                                                   2     /     /                                             2          2   \\
    4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|     4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|       4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|     4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|
1 - \/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *cos|-------------------------------------------------------------------| - I*\/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *sin|-------------------------------------------------------------------| + 1 + \/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *cos|-------------------------------------------------------------------| + I*\/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *sin|-------------------------------------------------------------------|
                                                                                 \                                 2                                 /                                                                                  \                                 2                                 /                                                                                    \                                 2                                 /                                                                                  \                                 2                                 /
(−i(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24sin⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)−(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24cos⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+1)+(i(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24sin⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24cos⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+1)\left(- i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} - \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + 1\right) + \left(i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + 1\right) 
=
2
22 
producto
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|       /                                                                   2     /     /                                             2          2   \\        /                                                                   2     /     /                                             2          2   \\| |       /                                                                   2     /     /                                             2          2   \\        /                                                                   2     /     /                                             2          2   \\|
|    4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|     4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|| |    4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/|     4 /                            2   /                   2          2   \      |atan2\-8*im(y) - 4*im(y)*re(y), -2 - 8*re(y) - 2*re (y) + 2*im (y)/||
|1 - \/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *cos|-------------------------------------------------------------------| - I*\/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *sin|-------------------------------------------------------------------||*|1 + \/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *cos|-------------------------------------------------------------------| + I*\/   (-8*im(y) - 4*im(y)*re(y))  + \-2 - 8*re(y) - 2*re (y) + 2*im (y)/  *sin|-------------------------------------------------------------------||
\                                                                                 \                                 2                                 /                                                                                  \                                 2                                 // \                                                                                 \                                 2                                 /                                                                                  \                                 2                                 //
(−i(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24sin⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)−(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24cos⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+1)(i(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24sin⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+(−4re⁡(y)im⁡(y)−8im⁡(y))2+(−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)24cos⁡(atan2⁡(−4re⁡(y)im⁡(y)−8im⁡(y),−2(re⁡(y))2−8re⁡(y)+2(im⁡(y))2−2)2)+1)\left(- i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} - \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + 1\right) \left(i \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + \sqrt[4]{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)}\right)^{2} + \left(- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{im}{\left(y\right)},- 2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 8 \operatorname{re}{\left(y\right)} + 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 2 \right)}}{2} \right)} + 1\right) 
=
        2          2                                           
3 - 2*im (y) + 2*re (y) + 8*re(y) + 8*I*im(y) + 4*I*im(y)*re(y)
2(re⁡(y))2+4ire⁡(y)im⁡(y)+8re⁡(y)−2(im⁡(y))2+8iim⁡(y)+32 \left(\operatorname{re}{\left(y\right)}\right)^{2} + 4 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 8 \operatorname{re}{\left(y\right)} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 i \operatorname{im}{\left(y\right)} + 3 
3 - 2*im(y)^2 + 2*re(y)^2 + 8*re(y) + 8*i*im(y) + 4*i*im(y)*re(y)
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