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x^2+2*y*x-y^2=-7 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 2            2     
x  + 2*y*x - y  = -7
−y2+(x2+x2y)=−7- y^{2} + \left(x^{2} + x 2 y\right) = -7
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
−y2+(x2+x2y)=−7- y^{2} + \left(x^{2} + x 2 y\right) = -7
en
(−y2+(x2+x2y))+7=0\left(- y^{2} + \left(x^{2} + x 2 y\right)\right) + 7 = 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:
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=2yb = 2 y
c=7−y2c = 7 - y^{2}
, entonces
D = b^2 - 4 * a * c = 

(2*y)^2 - 4 * (1) * (7 - y^2) = -28 + 8*y^2

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

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

o
x1=−y+8y2−282x_{1} = - y + \frac{\sqrt{8 y^{2} - 28}}{2}
x2=−y−8y2−282x_{2} = - y - \frac{\sqrt{8 y^{2} - 28}}{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=2yp = 2 y
q=caq = \frac{c}{a}
q=7−y2q = 7 - y^{2}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−2yx_{1} + x_{2} = - 2 y
x1x2=7−y2x_{1} x_{2} = 7 - y^{2}
Gráfica
Respuesta rápida [src] 
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                |            /                           2                        /     /                        2          2   \\|      /                           2                        /     /                        2          2   \\
                |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||   4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|
x1 = -re(y) + I*|-im(y) - \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------|| - \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *cos|----------------------------------------------|
                \                                                                 \                      2                       //                                                           \                      2                       /
x1=i(−(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))−(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−re⁡(y)x_{1} = i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} 
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                |            /                           2                        /     /                        2          2   \\|      /                           2                        /     /                        2          2   \\
                |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||   4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|
x2 = -re(y) + I*|-im(y) + \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------|| + \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *cos|----------------------------------------------|
                \                                                                 \                      2                       //                                                           \                      2                       /
x2=i((2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))+(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−re⁡(y)x_{2} = i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} 
x2 = i*(((2*re(y)^2 - 2*im(y)^2 - 7)^2 + 16*re(y)^2*im(y)^2)^(1/4)*sin(atan2(4*re(y)*im(y, 2*re(y)^2 - 2*im(y)^2 - 7)/2) - im(y)) + ((2*re(y)^2 - 2*im(y)^2 - 7)^2 + 16*re(y)^2*im(y)^2)^(1/4)*cos(atan2(4*re(y)*im(y), 2*re(y)^2 - 2*im(y)^2 - 7)/2) - re(y))
Suma y producto de raíces [src] 
suma
           /             ________________________________________________                                                    \       ________________________________________________                                                                  /             ________________________________________________                                                    \       ________________________________________________                                                    
           |            /                           2                        /     /                        2          2   \\|      /                           2                        /     /                        2          2   \\              |            /                           2                        /     /                        2          2   \\|      /                           2                        /     /                        2          2   \\
           |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||   4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|              |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||   4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|
-re(y) + I*|-im(y) - \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------|| - \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *cos|----------------------------------------------| + -re(y) + I*|-im(y) + \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------|| + \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *cos|----------------------------------------------|
           \                                                                 \                      2                       //                                                           \                      2                       /              \                                                                 \                      2                       //                                                           \                      2                       /
(i(−(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))−(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−re⁡(y))+(i((2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))+(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−re⁡(y))\left(i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right) + \left(i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right) 
=
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             |            /                           2                        /     /                        2          2   \\|     |            /                           2                        /     /                        2          2   \\|
             |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||     |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||
-2*re(y) + I*|-im(y) + \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------|| + I*|-im(y) - \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------||
             \                                                                 \                      2                       //     \                                                                 \                      2                       //
i(−(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))+i((2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))−2re⁡(y)i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - 2 \operatorname{re}{\left(y\right)} 
producto
/           /             ________________________________________________                                                    \       ________________________________________________                                                    \ /           /             ________________________________________________                                                    \       ________________________________________________                                                    \
|           |            /                           2                        /     /                        2          2   \\|      /                           2                        /     /                        2          2   \\| |           |            /                           2                        /     /                        2          2   \\|      /                           2                        /     /                        2          2   \\|
|           |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||   4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/|| |           |         4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||   4 /  /         2          2   \         2      2        |atan2\4*im(y)*re(y), -7 - 2*im (y) + 2*re (y)/||
|-re(y) + I*|-im(y) - \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------|| - \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *cos|----------------------------------------------||*|-re(y) + I*|-im(y) + \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *sin|----------------------------------------------|| + \/   \-7 - 2*im (y) + 2*re (y)/  + 16*im (y)*re (y) *cos|----------------------------------------------||
\           \                                                                 \                      2                       //                                                           \                      2                       // \           \                                                                 \                      2                       //                                                           \                      2                       //
(i(−(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))−(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−re⁡(y))(i((2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−im⁡(y))+(2(re⁡(y))2−2(im⁡(y))2−7)2+16(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(4re⁡(y)im⁡(y),2(re⁡(y))2−2(im⁡(y))2−7)2)−re⁡(y))\left(i \left(- \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right) \left(i \left(\sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + \sqrt[4]{\left(2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7\right)^{2} + 16 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},2 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 7 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)}\right) 
=
      2        2                     
7 + im (y) - re (y) - 2*I*im(y)*re(y)
−(re⁡(y))2−2ire⁡(y)im⁡(y)+(im⁡(y))2+7- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 7 
7 + im(y)^2 - re(y)^2 - 2*i*im(y)*re(y)
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